Professor X is easily explained (alternate universes), Emma Frost I already explained (again, alternated universes), Beast is easily explained (All mutants are still mutating. Rogue eventually go to the point where she CAN turn it off), The Summers brothers.... okay, you got me (But again... alternate universes), Sabertooth is easily explained (he went through a SEVERE mutation that wiped part of his memories... either that or by then he's so animal he doesn't care about brotherly love. By the by... remember Sabretooth's obsession with Wolvy's dog tags?), Moira MacTaggart (No one mentioned her age in EITHER movie and no one said she was american in First Class. That could've been a fake accent. Also alternate universes), General lies (Really? You REALLY gonna make me say it again? Alright. Alternate... Fucking... Universes), Deadpool... FUCK FAKEPOOL (UNLESS!!!!! They use him as a stepping stone for the REAL Deadpool. Check it. His mouth is unsealed in the end credits scene anyways (where he breaks the fourth wall, by the way), the decapitation fucked up his laser vision and technopathy (it's how Stryker controlled him) so that's gone, but it also glitched his teleport, so that it's not working properly... exactly like Deadpool's teleportation device. The torture he had to go through made him batshit crazy, so yay. And now he's ugly, so he wears a mask and kick-ass costume. Oh, yeah. And because of his love for his katanas, he pulls those fuckers out and puts them back into handles. BOOM!! Fakepool is now Deadpool, free to kill and screw hot chicks and kill some more as much as he wants).
There you go. Every problem fixed by simply saying "X-Men: First Class begins a separete Universe and Timeline that will be continued in the past segments of X-Men: Days of Future Past.
I love my brain sometimes.