X-Men: First Class/WMG

In the movieverse, Emma Frost is Cyclops' mother
Well, she'd be way too old to be his girlfriend. And it kind of fits with the decision to make Havok his father instead of his younger brother. On top of that, it would be kind of amusing to see everyone's favorite snarky, white lingerie-clad Dominatrix become the mother of the guy that she's currently sleeping with. Because, y'know...if you're going to annoy fans by messing with Canon, you might as well have some fun with it.

Darwin is alive.
He's Darwin, he can survive being atomized by a condensed plasma blast! He just needs some time to put himself back together.
 * In the comics, he did survive having his body destroyed. He grew a new one.

Fill in the continuity holes here.
The movie has too many similarities to the other films to be a reboot, so what happened was...
 * Charles regained his ability to walk before his appearance in Wolverine and The Last Stand's prologue, but lost it again before the original movie.
 * Erik occasionally found himself on good enough terms with Charles to work with him, such as when they build the second Cerebro.
 * Hank's cameo in X2 was either old footage, an actor hired by the real Hank to make public appearances, or the result of a temporary cure.
 * The Emma Frost that appeared in Wolverine was a separate character from the Frost in this movie, perhaps her daughter.
 * The young "Storm" and "Cyclops" in this movie were actually just similar-looking mutants.
 * The reason Xavier was surprised by Erik's helmet in the first movie is because the original helmet had been lost for years at that point.
 * The incorrect ages given by Charles in the first movie are the result of psychic tampering with his memories (perhaps by this movie's Frost, or even by himself).

Lets pretend some legal troubles have vanished for a second.
I'm going to preface this WMG by reminding us that it is very, very unlikely due to film rights issues that we all know and love (Or can google and hate). So lets pretend Marvel Studios still owned the film rights to X-Men and that this movie was in continuity with the MCU. In that case I would venture a guess that the helmet that was given to Shaw by the Russians was not built or designed by them but instead found in an archeological dig. To be more specific the helmet was found in a ditch with a bunch of very large human shaped packs of ice and a single seemingly human skeleton that was wearing some shiny armor and holding a sword with the word "Baldur" written on it in norse runes. The helmet was merely given to Shaw because thems russkies knew he liked fancy stuff and he discovered its ability to block telepathy on accident later and lied about its origins because it reminded him of his old rival Johan Schmidt. Other fun ways we could tie first class into the MCU? List below if you want.
 * Mystique's slowed aging could be used as the basis to introduce the infinity formula. Which then fell into Shields hands after Magneto broke into the CIA. Possibly using some on himself meaning they wouldn't have to recast Michael Fassbender for modern stories.