Has missing information been provided for the time between the Veronica Mars TV series and the movie?

Summary Answer:

Why didn't she join the FBI?
It would be hard to create a movie with all the former cast if she was in a totally different environment.

Who filmed the sex tape?
This appears to have been resolved in the TV series.

What happened in the Sheriff elections?
Keith Mars lost.

Detailed Answer:

I can try and answer some of these questions.

Why didn't she join the FBI?

From an interview panel involving the show's creator, Rob Thomas:

On the film's original plot:

The cast is very cognizant of the fact that the film would not have been made without the money donated by the fans, and Thomas said that once he knew it'd be financed that way, his original idea, which involved Veronica as an FBI agent, would no longer work. "I sort of abandoned that FBI idea and tried to find a plot to get back to all the characters we know and love," said Thomas. "Because if the fans are going to pay for a movie, I want them to see the characters they love."

In other words, if Veronica were a rookie FBI agent, she would obviously be in a different geographical location, well away from all the former characters. To prevent this possibility, the movie takes place primarily over an old high school reunion so many of the characters can be reunited.

From an interesting article on FlavorWire:

I could see how [the FBI idea] would have worked as a successful season of Veronica Mars, and definitely one that I, and plenty of fans, would watch and enjoy. It was a smart move to reset the setting and characters, because that would make the show more accessible to new viewers — which is exactly what The CW was looking for. The FBI angle turns it into more of a crime drama with a whip-smart lead, rather than a teen drama with a sassy lead (because, unfortunately, teen dramas will always get a bad rep but crime shows will always be popular).

In fact, that’s where the Season 4 pitch wins over the movie: Veronica as an FBI agent makes more sense than Veronica as a lawyer. And sure, the movie addresses the former career option and, yes, Veronica would make a hell of a lawyer (remember the “One Angry Veronica” episode?), but it’s not her. She needs to be constantly on the move, solving cases, and in the mud. Thomas seems to agree; a few months prior to starting the Kickstarter, he was still intent on having the film center on Veronica in the FBI. Once the show graduated from high school, it would naturally become more about the mysteries than the setting.

But the Veronica Mars movie was fan service, and that’s why it worked so well. The fans are the reason it happened, so Thomas had to abandon his original idea because there was no way to do it while bringing back all of the favorites from Neptune. The movie was packed with callbacks and inside jokes — there is even a clever exchange about this pilot pitch, referencing Veronica’s FBI career as something that happened in another life — because, essentially, that’s what fans paid for. The movie wasn’t much more than a long episode of the show, but that’s exactly what I loved about it. Yet I also loved where the show was going to go in the season-that-never-happened. It would have been great to see a longer version of Veronica adjusting to the FBI and generally just kicking ass and shooting bad guys. And maybe we’ll get some stories of her in the law enforcement eventually (there are rumors of a second movie, and Thomas hasn’t been shy about his desire for a Netflix deal) but for now, Veronica’s intriguing alternate life will remain too brief.

I think the logic applied here is completely rational. It just wouldn't have worked to have many of the old cast back and Veronica in the FBI.

Who filmed the sex tape?

The Washington Post did a recap of the series finale before the movie, commenting:

The series finale seemed more like a season finale. It was unsatisfactorily open-ended and left many unresolved issues. Last we saw, Veronica she was a freshman studying criminology at Hearst College in California. A sex tape of her and boyfriend Stosh “Piz” Piznarski (Chris Lowell) mysteriously surfaced. Neptune’s resident bad boy and Veronica’s ex-boyfriend Logan Echolls (Jason Dohring) assumed Piz was the culprit and beat him to a pulp.

Once it was revealed that a secret collegiate society (modeled after Yale’s Skull and Bones) was actually behind the tape, Veronica became furious at Logan. She tells Logan their friendship is over and wants nothing to do with him, thus seemingly choosing the sweet, levelheaded Piz over her former, long-time flame (continuing an epic love triangle that picks up in the movie).

Vengeful, Veronica hunts down the secret society and stages a break-in at a billionaire’s mansion to collect evidence on the association’s members. While escaping the residence, she is caught by security cameras. Her father, former private eye and current sheriff Keith Mars, protects her identity by destroying the surveillance footage and, by doing so, jeopardizes his re-election campaign. The final scene shows Veronica casting a ballot for her dad in the sheriff election and walking off alone, seemingly debating whether her sleuthing has caused more harm than good.

I never finished the series myself, but this sounds like that thread was fairly well resolved in the TV series. I'm not aware of any mention of it in the movie.

Who won the sheriff's election?

The Veronica Mars Confessions blog stated:

Note: Rob Thomas tweeted from his official Twitter (@RobThomas) that Keith Mars lost the Sheriff election to Vinny Van Lowe. He also had plans for The Castle and Gory Sorokin. As for Logan and Veronica, he refused to answer the question. As for the movie, he said “Don’t give up.”

Unfortunately I can't find the direct tweet in question, but as I've seen this claim in a few places I assume it is true.

ScreenRant also had an interview with Rob Thomas, where he stated:

“She transferred away from Hearst college after that freshman year and went up to Stanford and finished her undergraduate degree. And then she went to Columbia Law School. She’s certainly been back to Neptune from time to time to see her dad and Mac and Wallace, but she hasn’t worked a PI case since the last one we all saw in the final episode of season 3. She ruined her dad’s chance for re-election and she decided that that path had caused too much chaos in her life.

“And so she hasn’t worked a PI case in nine years. And when we meet her, she’s sort of like where Tom Cruise was in the beginning of The Firm. She’s between having graduated and taken the bar and she’s interviewing at various law firms in New York.”

Final Thoughts

In the same interview as the last quotation, the following exchange occurred:

Will the movie be new audience-friendly? According to Thomas:

“I want to avoid getting deep into the ‘Veronica Mars’ mythology. We’re not going to get ghost visits from Lilly, as much as I joked about having Amanda Seyfried in it. We’re not going to get into Veronica’s backlog of cases. I want to make the movie accessible for people who haven’t watched ‘Veronica Mars’ before. All you need to know is that Veronica was a teenage private eye and she’s given it up. And there will be a moment early in the movie that will bring her back to Neptune and make her give being a private detective one more shot.”

Therefore, it does seem like Rob Thomas was trying to target new viewers as well as the old fans. It definitely appears logical then that some of the loose ends were simply left, as it would be too convoluted for new viewers to understand them. Other threads were resolved in the movie and most other discussion points appear to have been addressed in interviews with Rob Thomas and the cast.

Did Veronica get an internship at the FBI? What happened to that, and why didn't she join the FBI?

This is addressed in the course of the film as an off-hand comment by one of the characters speaking to Veronica, summarized as, "Someone told me you joined the FBI.", which Veronica dismisses outright as a false rumor. This was an intentional nod to the proposed FBI season.