I wonder what would happen to the primary if, however unlikely it is, Trump were to not only be impeached, but removed. Would he still be able to run for a second term? It doesn't look as astronomically unlikely as it used to that the Senate could convict, if the Republicans are actually that pissed about the Syria situation and the polls support it enough, which they're already at majority support among Dems and Independents so it's getting close, we're at Nixon levels of support for it now. If that happens, does the Republican primary begin in full force with Pence as the incumbent? Or does Trump remain at the top of the ticket, with Pence as VP? Or would Trump still be able to run, but would Pence also be a candidate, and Trump would have to choose a new VP? If Pence and Trump were both running, with Pence as the technical incumbent, who would the RNC support? Would they still support them enough to cancel primaries? Seriously, I get that this is unlikely, but I wish someone with more knowledge on the laws behind it would delve into this possibility because I'm really curious how this would work.
These are great questions and I'll try to answer what I know (or suspect) line by line.
Would he still be able to run for a second term?
This depends on the verdict returned by the Senate. There is removal from office and there is "disqualified" from office. If he is only removed, he can technically run again because he's not exhausted his constitutionally limited "elected" 2 terms in office. If he is disqualified from future office, then the obviously cannot run again. Keep in mind that even if he is only removed from office, the Republican Party may not accept him as their nominee because the House could immediately refile Articles of Impeachment if he is re-elected and just remove him from office again.
If that happens, does the Republican primary begin in full force with Pence as the incumbent?
No, Pence only maintains office for the remainder of the term. To be considered for election, Pence would have to submit his candidacy for POTUS documents to the Federal Elections Committee and the Republican Party would have to be willing to accept him over the currently running candidates....which could cause a ton of friction within the party depending on when the final verdict of impeachment happens - before or after the Republican National Convention.
Or does Trump remain at the top of the ticket, with Pence as VP?
See above first answer. Trump can still be the nominee if he is only removed but not disqualified. But even being removed pretty much locks him from being the nominee again. Pence would be the VP nominee with him again so long as Trump selects him for it.
Or would Trump still be able to run, but would Pence also be a candidate, and Trump would have to choose a new VP?
Technically, Pence could announce his candidacy now and run against Trump and the others but it would preclude from being Trump's VP. Meaning Trump would have to pick someone else for VP (Graham?).
If Pence and Trump were both running, with Pence as the technical incumbent, who would the RNC support?
Technically, the Republican Party can select whomever they want as their candidate (as long as it doesn't violate FEC rules).
If Trump is only removed from office but not disqualified before the RNC, Trump would still likely be their candidate.
If Trump is only removed from office but not disqualified after the RNC, the candidate should be whoever won second place for the nomination but party has every right to over-ride the process and select Trump or Pence. Still likely to be Trump.
If Trump is disqualified before the the RNC, Pence could become the nominee if he registers in time.
If Trump is disqualified after the RNC, the candidate should be whoever won second place for the nomination at the RNC but again the party can select Pence if they want regardless of RNC results.
Would they still support them enough to cancel primaries?*
No matter when the final impeachment verdict arrives, it will likely be too late for the currently canceled primaries to happen. Nor will it likely matter since the party can basically select who they want. It doesn't have to be the democratic process we have now. In fact, these open selection processes have only been around since the late 60s to early 70s. Before then, the parties just picked who they wanted. With that said, the currently canceled primaries will likely stay canceled.
* Primary vs Caucus. A primary is a state managed selection process. States fund and manage them. A caucus is party managed selection process. The party funds and manages them. With this in mind, states that only hold a caucus will already cancel them if the party has a candidate they want. If a state that thinks a primary is an unwarranted expense (incumbent president or overall clear candidate), they will cancel them too.