What is the relationship between theaters and film distribution?

There are no "end to end" companies. United Artists cinema chain is actually part of the Regal Entertainment group, while United Artists the motion picture company is part of the MGM holdings group. As far as I am aware (and have seen in a quick look) no company also owns a theater chain.

Theaters "can" show any movie that they can get a contract for. However, smaller or older theaters may not be able to secure contracts for some of the large budget mainstream releases, as they cannot guarantee enough revenue in gross tickets sales to get the rights to show a movie. Other houses brand themselves as "indie" and get independent or small run films (Cartoon shorts, short films, documentaries, etc.)

When a movie is created, it is sent back to the studio, who makes a licensing agreement with a distribution company. In some cases (Such as Paramount, MGM and a few others), the film studio is also a distribution company. In other cases, the distribution company is a separate entity. The distribution company is in charge of determining how many copies to make, and showing (screening) the movie to prospective buyers for the various theater chains, and sets a release date.

Once it is screened, if a theater wants to acquire the movie for performance, they make a contract with the distribution company outlining how many screens it will be shown on, percentage of gross ticket sales, when "passes" and other special items can be used (Paramount is notorious for never offering any kind of discount/pass during the initial few weeks of release).

The distribution company is in charge of getting the film to the studios a few days before release, monitoring the length of the showings (called the engagement), collecting revenues and retrieving the film copies after the showing. They are also responsible for determining the type of release, whether it's theatrical, video on demand (VOD), straight to DVD, etc.

There are companies that handle all releases, there are others that only handle home distribution, or DVD/Blu-ray distribution, and they may or may not be the same entity as the one that did the theatrical release.

Typically releases are timed so as not to overlap, and the "typical" viewing window for a theater release is 17 weeks, at which time the DVD is released. Originally it was around 6 months, but studios have been pressing for shorter and shorter theater runs, to get more out of DVD sales which continue to decline. Obviously, theater owners resist this as they depend on the ticket sales (and the concurrent food sales) to generate their revenue.

This is the general model for the bigger studios, who purchase film scripts and rights and produce. For others, they create the films and then "shop" them around to companies and distributors for interest, and/or submit them for consideration for film festivals (Sundance, Cannes, etc). They may also try to secure their own distribution contracts, but that is a broader aspect that isn't really related to this question.

It used to be end to end before the Consent Decree in 1948 against Paramount Pictures, and by implication, all the studios. Before that, Paramount made the picture, and told which of the theaters it owned, when to play it, for how long, and how much they'd get as revenue. In part, that practice included 'block booking', now also no longer permitted. That meant a studio could tell a theatre, "If you want that good picture we weren't going to give you to play, then you have to take these six bad ones you don't want."

Now production and exhibition are two different businesses, owned by different parties. Nothing in 2016 looks like it did in 1947. Now content producers worry their customers will watch content for free, on the web & their devices. Disney's stock took a hit when the CEO announced declining ESPN viewership.

Typically, the percentage of box office receipts that go to the distributor is very high for the first few weeks of the movie's run. So, for example, Big Movie opens and for the first two weeks, 90% of box office receipts go to the distributor. Most distribution deals offer better percentages to the theater the longer the film plays. So maybe week 3 it's 75% to the distributor, 25% to the theater. Maybe week 4 is 65%/35%.

But really it's between the theater booker and the distributor as to what kind of deal they work out.

Ideally, a theater's take of the box office receipts would pay all the utilities and maybe some of the payroll. Then the net profit from the concessions is where can they really make their money.