Honestly, I know they want to do 90s, but I say keep it in the 80s. Following X-Men: Apocalypse, Fox have the opportunity to do something they really haven't had since the very first X-Men, showing Xavier's School as a school. Yeah in the first three we see students, we see teaching and to a much lesser extent, we see that in this film as well. Just like Spider-Man: Homecoming will give us a pure unadulterated film of Peter Parker as a high schooler, the next two or three X-Men should do the same for Jean, Scott and Storm.
I know some people are like they should do catchup to the present because Days of Future Past's ending told us what happens, but when I see that, I think, so what? We know what happens either way, a happy ending. Only difference is that this time actually know what it looks like. There's like thirty-forty years of history between those ends and I think putting a focus on the years that happen after Apocalypse battle in 1983, showing Scott, Jean and Storm grow up and become teachers from students (instead of skipping a decade again) would be the best.
That's just my two cents though. Funny, I was actually going to come here and create a thread on why the series should stay in the 1980s but this seems like a good fit for that discussion.
Going by your questions, though, what would I do for the next X-Men Film:
When is it set?
* I'd say 1983, months after X-Men: Apocalypse:
Tied to any particular historic event?
* Doesn't need to be, deals with the aftermath of the Apocalypse event.
Where are the characters when it opens and how would it progress?
* Focus on Professor X guiding the X-Men, which consists of Beast, Cyclops, Phoenix and Nightcrawler. Start the movie off with them performing a rescue op against Essex Corporation by night, roll intro credits, then show them as students by day. I don't even know if introducing Mr. Sinister would be the best move here, but show the presence that he has. Include new mutants like Frederick Slade, or Ozymandias could make good villains. For plot, I'd try and make it revolve around Apocalypse's armour - how did he have access to those designs and all? It would be a good way to tie it in to the Shi'ar. Essex wants the Shi'ar technology that Apocalypse gained access to, so Ozymandias and Frederick Slade (or any other evil mutant really) work against the X-Men to get it. Show the budding romance between Jean and Scott, watch it blossom as they both learn to control their abilities.
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