Quicksilver for you (the guy who can run fast), yeah they could have used him. In the comiocs timeline, he originally became one of the "Brotherhood of Evil Mutants" and was with Magneto, along with his sister "the Scarlet Witch". Both were originally thought to be Magneto's children and he claimed that was the case, but it was found later that they had been saved by him as young children. They've obviously changed quite a bit for the X-Men movies, bearing in mind he was an American kid called Peter in this film.
This was set in late 60's/early 70's, but Quicksilver and the Scarlet Witch are both in the next Avengers movie, though being the older Quicksilver he's going to be played by a different actor and listed as Pietro Maximoff, so there's potentially a major issue there.
A lot of those plot holes were because they're trying to align to two series of films, because Frist Class was supposed to be a reboot, but it left them with the problkem of integrating some of the characters from the original movies as most people are quite happy with Hugh Jackman as Wolverine. The problem I can see now is how thye are going to keep McAvoy and Fassbender in their roles because it showed Patrick Stewart as Xavier at the end, so presumably the next film will have him as Xavier. Though with it being Apocalypse, a virtually immortal mutant from ancient Egypt, the story could also span timelines and have both old and new Xavier and Magneto. I still think it's going to be a problem for them.
A lot of what you mention wasn't a plot hole at all. They did try and close a lot of the plot holes, time travel and altering history is always a good tool for that.
The Days of Future past bit altered the timeline somewhat after the first film, so effectively what came to pass in X-Men 2 and Probably the Last Stand (with Prof X and Cyclops popping their clogs, hasn't happened. Days of future past ended where the first X-Men film started. The reason they killed them both off in Last Stand is because Stewart and Marsden both said they didn't want to do the roles anymore. Obviously money talks.
I liked how they integrated the Wolverine memory loss and how he was passed on to the people who gave him his adamantium skeleton eventually.
The Kitty Pryde thing, Shadowcat, is a bit strange. A lot of mutant's powers have evolved over time in the comic books, but I can't remember if this happened or not, or might have happened after I'd stopped reading them. The actual Days of Future Past was quite a long story arc in the comics.
As to the Mystique/Sentinels business. Well the sentinel program has been a recurring theme throughout the X-Men timeline. They'd beaten the sentinels a number of times, with Sentinels becoming more advanced all the time, but sentinels killed mutants, it's only natural that a serum to strip them of their powers would be a preferred option for many.
As to Mystiques capture, that one I'll have a think about, but my biggest issue with that part of the story was the fact that they were integrating her DNA into synthetic lifeforms and, not only that, it enabled those advanced sentinels to mimic the X-Men's powers, something she couldn't do, only shapeshift and her powers have never evolved in that way as far as I'm aware.
Now if it had been Rogue (Anna Paquin) then she could take other mutants powers temporarily, but still integrating something biological like that into a sentinal, well it's hardly surprising they didn't try to explain how they did it.
Going back to Xavier's "death" and Magneto's powers (I've just stuck the DVD on and skipped to the end because I remembered something about this.
The bit at the end of the Last Stand (after the credits) alluded to the fact that Charles Xavier's conscousness at least was still alive. It was a scene in Moira McTaggart's lab. A body in a bed with its face covered says "Hello" Moira", to which she replies "Charles ... " Then the film ends. Which clearly left it open for Xavier to return and for it to be a different actor if necessary.
The very last scene
before the end credits of Last Stand showed Magneto moving a chess piece slightly in a park (just after Angel flies over), so his powers were already coming back. Any other mutants who lost their powers probably got theirs back as well. The future scenes were a fair few years after the Last Stand, so he'd had plenty of time for his power to fully re-establish itself. It was never going to be permanent. It also might explain why the sentinel program got restarted.
Oh and just found this on IMDB about Xavier being alive.
In the mid-credits scene of The Wolverine (2013), Xavier is revealed to be alive and kicking in his old wheelchair bound body. With Wolverine wondering how this is possible, Xavier simply responds with "As I told you a long time ago Logan, you're not the only one with gifts".
Although a full explanation is absent from both The Wolverine and X-Men: Days Of Future Past, there are two different theories uttered by the producers of the franchise about how all this is possible. The first is in the dvd commentary for X-Men: The Last Stand, where it is theorized that the brain dead patient was in fact Charles Xavier's identical twin brother. This borther was born without any brain activity due to Charles' overpoweringly strong brain in the womb. The second was given by X-Men: Days Of Future Past screenwriter Simon Kinberg in an interview. He stated that dialogue was written for the 2023 scenes that explained how Xavier's old body was reconstituted by an unnamed mutant to its old and familiar state, allowing Xavier's consciousness to transfer back to its own body. However, the dialogue didn't make the cut as it would clutter the film's already complicated plot with even more explanations.
None of the two theories have been confirmed to be canon, which leaves the answer to this question open to the interpretation of the viewer.