What rules govern how TV show opening credits are structured?

The rules for how the order of credits is produced is very long and convoluted. They also vary from show to show, so for any one "rule of thumb" it's almost guaranteed you can find an exception on the air right now on some channel.

The basic guidelines that most shows follow most of the time are drawn from the rules for the various industry guilds. In the case of actors, that would be SAG (the WGA and DGA would determine how/when a writer or director get credited.) The Wikipedia articles linked in a few comments/answers gives a basic rundown of all the guild rules that have to be followed in the "typical" credits sequence. Note that movies opening and closing credit sequences differ a lot from television ones (in particular, you rarely see the name of the production company in a TV show's opening credits.)

However, there's a lot of variation on the basic theme, because the placement and styling in the credits sequence is often part of an actor's individual contract with the studio. Studios will rearrange the credits or give certain actors special consideration in the credits as part of the negotiations.

The basic flow of credits is as you mentioned: core actors first, followed by guest actors, followed by a handful of top-level production staff. In TV, it's rare to have a full set of closing credits the way you have in movies, where the entire cast would appear with their character names, for simple time reasons: they would take a really long time that could instead be used for commercials. (In fact, many networks now show split-screen closing credits side by side with a promo for the upcoming program.) So those credits are usually limited to production-level people and companies, copyright/legal notices, and other minutia.

Why a particular actor gets a particular spot in the credits is mostly a matter of PR, and explains some of the variations you mention in your question:

• Being billed before the title is "more prestigious" than after the title; usually only the one or two top-tier stars will get this billing, if at all. In ensemble shows it's rare to see this type of split billing.
• Actors are generally billed in rough "order of importance" to the show.
• Being billed last, however, is also considered a key position, as long as it is somehow differentiated from the rest of the cast. For example, in the Buffy TV show, Anthony Stewart Head (arguably the most "famous" cast member prior to the show) was billed as "and Anthony Stewart Head as Giles" - the only cast member whose character was named.
• Since the opening credits are pre-recorded, typically only season regulars make it into the "core" credits sequence (think "theme music") for shows that have them. Recurring guest stars are usually billed overtop of the opening scene, introduced with "Also Starring".
• Single-episode guest stars go last; depending on how famous/important they are, they may also get the "as charactername" treatment.

In general, having a unique style of credit (e.g. being specifically associated with a character) is a bonus given out to special guest stars and particularly important actors that lets them stand out from the rest of the cast. This is something a well-known or highly-priced actor would negotiate as a condition of their appearance. They even negotiate down to the exact wording, which can include "guest" vs. "special guest", are they "starring" or "appearing", etc (I have seen shows that had multiple guest stars, each billed slightly differently).

Another key distinction to make is between series regulars and recurring guest stars. A series regular will still appear in the opening credits, and be given credit for the episode, even if they don't actually appear in the episode.

The general idea here is that a series regular, even if they aren't in a particular episode, is contractually required to be available to appear in that episode. A recurring guest actor, on the other hand, is typically contracted to appear in only a specific number of episodes, and can usually skip a particular episode if they have prior commitments. A recurring guest star, with lower billing, may actually get more screen time than a series regular that gets top billing. This happened in House in seasons 4-6, where Olivia Wilde, Kal Penn and Peter Jacobson were given lower "Also Starring" billing, while rarely-appearing Jennifer Morrison and Jesse Spencer continued to be billed as series regulars. As always, this was entirely down to the contracts the latter two had with the show.

It seems to be an amalgamation of agreements between the various guilds: the screen actors guilds, the screen writers guild, and directors guild. These aren't set in stone and there are many different variations to opening credits. Wikipedia has a pretty good article on the subject.

Opening Credits on Wikipedia

It's not only a prestige thing, to get listed in the credits, it also has to do with money:

the WGA Agreement requires that the writer or writers accorded 'created by' credit on a series receive a royalty (or payment) for each episode of the series that is produced beyond the pilot.

-https://www.writersstore.com/the-ins-and-outs-of-tv-series-writer-deals/

This boost is somewhere between $1'000 and $ 6'000 per additional episode.

It is mostly a contractual issue, so there is not much sense to make out of it. Oftentimes an actor will negotiate with the producer specifically about how big their name appears and in what order in the credits.

The most vivid example which comes to mind is Andy Kaufman in Taxi getting a lead credit, which was negotiated by his talent agent. The credit was very weird considering his small role in the series and obscurity at the time.

Opening credits are a functional as well as creative part of movies. Telling the audience who is in the movie, as well as other known talent such as writers, directors and composers, sets the stage for what’s to come. Check out an example here.

With a hundred years of moviemaking history, there must be a “standard” way of listing people in the opening credits, right? Wrong. Except for the contractual or union-mandated billing, the rest of the opening credits have no rules.

Many great movies had no opening credits, such as The Godfather and Apocalypse Now. Citizen Kane opens with only the title of the movie, saving all the credits for the end.

Most movies in the 1930s through 1950s listed lots of cast and crew in the opening credits, often in a sort of Playbill format as if the movie was a filmed stage play. Today, high-powered talent demand specific types of onscreen billing in their contracts.

Although it’s common today for blockbuster movies to open with just the title and no other front credits, it is reported that George Lucas was fined $250,000 by the Directors Guild of America for not putting the director’s name (his) in the opening credits.

Independent, non-union movies aren’t bound by union rules for opening credits, so anything goes. However, over the past two decades an order of listing cast and crew has become widely accepted by Hollywood studio and independent productions.

When the lights dim and the movie begins, the opening credits will be presented in the following manner:

STUDIO OR DISTRIBUTION COMPANY presents

This is the distributor. It may be a studio or independent distribution company. It may or may not have financed the making of the movie. But it is the company that sold the movie into theaters, television, DVD, etc. Sometimes it is listed in the same text as the rest of the credits. Or it could be a logo or even a standalone clip, such as the MGM lion roaring.

THE PRODUCTION COMPANY

Usually a movie is produced under a business entity that finances motion pictures. It may be an independent company, a studio or a subsidiary. The director or producer’s personal production company may also have acquired the source material, such as a book or play, upon which the movie is based, and could get a production company credit here as well.

IN ASSOCIATION WITH SO-AND-SO PRODUCTION COMPANY

There may be multiple production companies that played a part in getting the movie made. They are listed here, after the distributor and the primary production company, studio or studio subsidiary.

A DIRECTOR’S NAME FILM (or Producer’s)

Audiences today are used to seeing the director’s name before the movie title, as the auteur or visionary of the movie. Some directors have developed such a well-known style that audiences will have specific expectations when seeing his or her name in front of the title. There’s a difference between a Tarantino film and a Michael Bay film.

If the director simply worked for hire without developing it from his early vision, it’s kind of silly to make the auteur claim here. Sometimes the executive producer claims the film’s style and vision. Tim Burton has with movies he produced but did not direct, such as Abraham Lincoln: Vampire Hunter and Batman Forever.

THE MOVIE STARS

Anywhere from one to three lead actors are often listed just before the title. It’s a similar position to the auteur, as the presence of these star actors is the reason many people came to see the movie.

True movie stars transcend their producers and directors. One may think of a Tarantino film, or a Spielberg movie or Coen Brothers picture. But more likely, people are there to see a Brad Pitt, George Clooney, Meryl Streep or Robert DeNiro movie, regardless of who produced and directed it.

Their names are listed before the movie’s title, which is equivalent to having their names above the title in the movie poster. This is where the term “top billing” originated.

THE TITLE

Now the title of the movie, by itself in big letters.

THE TOP CAST

After the title, the leading actors are presented, either with their names singly on their own “title cards,” or sometimes in twos or threes on the screen together, if the actors have more or less the same size parts or are of equal status.

SUPPORTING CAST

Other actors get listed next, if they have significant parts (several scenes) or they are recognizable names. Often several actors will be listed together on the same title card.

THE “WITH” ACTOR

An actor or actress of some renown may have a small part in the movie. He or she might get a “with” credit here, such as “with Margot Thespian.”

Often these credits go to actors who “used to be” somebody years ago, still have a recognizable name, but are no longer box office draws.

THE “AND” ACTOR

A step up from the “with” acting credit, the final title card for the cast is reserved for the actor who has a meaty part in the movie, but not a leading role. It’s like, wow, isn’t it amazing that Big Actor blessed this movie with his presence, and I bet he’s gonna be great!

It’s where the elder statesman is listed, the uber-star. Many times the character’s name is listed here as well, such as “And Sam Elliott as The Stranger” from The Big Lebowski

That’s the production companies and the cast. Now for the crew.

CASTING DIRECTOR

Who brings all the great acting talent together for this movie? Sometimes the producers, sometimes powerful talent agencies package their clients into a movie. More likely a casting director puts the cast together, if not the lead actors then all the supporting players.

Casting Directors can be members of the Teamsters union and may also belong to a professional organization such as CSA, the Casting Society of America.

Mix-n-Match the following…

The order of listings in the next section of production people is less rigid. Some movies with big special effects might put the Visual Effects Supervisor first, followed by the others in this section. Some movies don’t list the VFX Supervisor until the end credits if the job was minimal.

Regardless of the order, most movies today include the following listings in their opening credits. They are usually listed singly on their own title cards, but not always.

MUSIC COMPOSER

May be listed as Music Composed by, or Original Score by, or even just Music by.

If the movie includes a vocal song, perhaps performed by a well-known artist or group, this is sometimes included as part of this title card.

PRODUCTION DESIGNER

The Production Designer is responsible for the overall physical look and feel of the movie. He or she works to achieve the director’s vision of the sets, locations, costumes, hair, makeup, special effects, colors and tone.

ART DIRECTOR

The Art Director oversees set design and construction, decorating or set dressing, props, signage and modifying locations. Sometimes the art director is pushed to the end credits.

SET DESIGN

If a movie makes use of lots of sets, as opposed to on-location shooting, the Set Designer may be listed in the opening credits, although this is sometimes another one of the credits that gets rolled at the end of the movie.

COSTUME DESIGNER

Someone designs the clothes for all the actors, makes or buys them, or at least guides the actors if they’re bringing their own modern-day wardrobe on low-budget productions. Certainly their work is evident in period or sci-fi movies, but even modern-day films need the skills of a talented costume designer to make the actors look good, and right for their parts.

HAIR / MAKE-UP ARTISTS

Movie stars are beautiful people, no one can argue that. They tone, tan, sculpt and starve their bodies to be the people everyone else in the world wants to look like. However, truth be told, to look their best on camera they need makeup and hair stylists.

Makeup is sometimes listed in the opening credits if the movie relies on makeup for special effects or advanced aging of a lead character. Think Planet of the Apes, Benjamin Button or even Forrest Gump.

Lots of stars have their own personal hair and makeup artists. When there are several of these, most of the time they are listed in the end credits and not in the opening.

SOUND RECORDING

The Sound Recordist is hardly ever noted in the opening credits anymore. Few producers or directors even know just what the sound people do, from recordists, mixers, editors to designers. So they usually get listed in the end credits, despite the fact that movies are a combination of picture AND sound.

VISUAL EFFECTS SUPERVISOR

It’s not just Harry Potter or Transformers movies that utilize visual effects, or VFX. A movie may contain dozens of VFX shots that the audience never notices, such as digitally changing license plates on cars or signs on walls, or making it look like the scene was shot in New York instead of on a street in Hollywood.

Or even digitally removing a pimple from an actor’s face that makeup couldn’t hide. These are all visual effects, along with the space ships, demons, monsters and fantasy worlds.

The VFX Supervisor will work on-set with the director and director of photography to help set up shots that will be digitally manipulated in post-production. During the post phase, the VFX Supervisor will oversee all the specialists who work on the digital shots, such as animators, compositors, rotoscopers and graphic artists.

EDITOR

The Editor takes the pieces of film or digital files and assembles them into the story that becomes the movie. He or she makes the editorial decisions about which shots to use, often in conjunction with the director, producer and studio executives.

DIRECTOR OF PHOTOGRAPHY

The DP heads up the camera crew. He or she will oversee the lighting, camera set ups, lens choices, filters, equipment and sometimes even the framing of all the shots, depending on the director. The DP is essentially 2nd in command on the set, beneath the director.

Sometimes the cinematographer is listed earlier in the credits. Typically he or she is listed just before the producers, writers and director.

EXECUTIVE PRODUCER

Producer titles have become extremely ambiguous. The Producers Guild of America has been trying to standardize the roles of the different types of producers, and even limit the number of producers eligible to be considered for Academy Awards.

Historically, the Executive Producer was the person who put the project in motion, either by acquiring the rights to a book or play or person’s life story, or by financing the production or bringing together the financing from investors or a studio.

The EP often does not participate in the day-to-day production of the movie, but may oversee Producers who do.

PRODUCER(S)

The Producer runs the operation of the movie — making hiring decisions, budget planning and expending, and supervising the cast and crew from pre-production to production and through post-production.

In addition to the Producer, there may also be Co-Producers, Line-Producers, Supervising Producers and Production Supervisors. These have defined roles (at least theoretically) by the Producers Guild, a trade organization.

Associate Producers are supposed to be sort of junior producers. Above an assistant, with some specific production responsibilities, but without full authority. However, in practice over the years, the Associate Producer credit has been handed out like candy as a perk to whoever the Producer wants or needs to give it. Girlfriends, mistresses, lazy nephews and entourage hangers-on have been named associate producers.

BASED ON THE BOOK (PLAY, GRAPHIC NOVEL etc.) BY

If the movie is based on another artistic work. This credit is often listed underneath and in smaller font size than the screenwriter.

STORY BY

Someone may have come up with the original story but didn’t write the screenplay, or wrote it with another writer. This credit acknowledges the contribution of the scenarist.

WRITTEN BY

The Writers Guild of America, West, dictates that the screenwriting credit shall be Written By, instead of Screenplay By.

There can only be up to three writers credited for the screenplay. However, a writing team is considered “one” credit, so in practice there can be more than three names in the Written By credit.

In the case of multiple writers working on the screenplay (at different times, usually), the names are listed chronologically from top to bottom. The original writer would be credited first, and below him or her the subsequent writers.

When teams are involved, the WGAw helps identify them through the use of “and” versus an ampersand. For example, if John Smith and Jane Doe worked together as a team, and then Frank Fellow was hired later to polish up the script, their credits would be listed as follows:

Written by John Smith & Jane Doe and Frank Fellow

The ampersand indicates that John and Jane worked as a team, while Frank wrote alone and on a subsequent draft from John and Jane’s.

DIRECTED BY

The Directors Guild of America (DGA) permits a movie to list only one director, even if two or more worked on it. A team is considered one directing credit, such as Directed by Joel and Ethan Coen.

Except in very rare cases, such as the death of a director during production, only one directing credit is listed.

The Wizard of Oz had five directors, including Richard Thorpe, the original who was fired after two weeks, George Cukor, who sort of babysat the production briefly until Victor Fleming was assigned. Fleming directed most of the movie, until he left to direct Gone With The Wind upon which King Vidor came aboard to finish, directing the Kansas scenes and Producer Mervyn LeRoy later directed some pick-up shots.

But only Victor Fleming got the directing credit. Because just like a ship can have only one captain and a kitchen one chef, a movie can (usually) have only one director.

And once the director’s name shows on screen, it’s time to start the movie.