Main X Men Films guesses here.
In the movieverse, Emma Frost is Cyclops' motherShe'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.
Xavier was always bald.That's why he told Beast not to touch his hair because he was really wearing a wig. Maybe he uses his powers to make people think that he has hair.
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. The comics support this idea; 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...
Lets pretend some legal troubles have vanished for a second.Let's pretend Marvel Studios still owned the film rights to X-Men and that this movie was in continuity with the MCU. In that case, 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.
Magneto lived in Ireland for a few years.Hence his having a noticable touch of an Irish accent - he learned to speak English in Ireland. It's also posible he was hunting Otto Skorzeny who was living in Ireland at the time.
X-Men movieverse is an Alternate History in which JFK wasn't killed.There are rumors that the X-Men: Days of Future Past movie will begin with Magneto killing JFK, explaining the bullet's weird trajectory. In the comics, Bad Future was the result of Senator Kelly being assassinated by the Brotherhood, but movie Kelly is too young to be important in the timeframe of the movie. So I think it will be like this: Magneto assassinates JFK, causing the Bad Future, which will have the actors from the original trilogy reprise different versions of their characters. In the end, JFK's death will be prevented using Time Travel, thus causing the X-Men movieverse as we know it.
The Man In Black is Fred Duncan.In some of the earlier X-Men issues, FBI agent Fred Duncan served as the team's government liasion. The Man In Black isn't an FBI agent, but he serves a similar role in the movie. In the sequels, he could serve as the team's spy in the CIA.
Shaw wasn't actually killed.The coin Magneto shoved through his head seems to have missed his medulla oblongata, the part responsible for life functions. Shaw was left alive, but seriously wounded (maybe for life), dependent mostly on the common humans he scorned.
Klaus Schmidt is related to Red SkullThe Red Skull from Captain America: The First Avenger to be specific. His name is Johan Schmidt and they share an interest in higher life forms than humans. They could be family, in-laws, or maybe part of a Evilutionary club whose members adopt the name "Schmidt". It's such a common name that it could be an ironic in-joke for the superpowered elitists.