When X-Men First Class was released, it created a few illogicalities in the movies-series continuity. I thought the problems trough and came up with a solution ideas. Some of this ideas I have mentioned before in my comments.
1)Prof. Xaviers walking ability
In FC Xavier was young, and had hair, when he lost his ability to walk, how ever we saw him walking when he was older and bald in the end of X-men origins:Wolverine and at the beginning of the X3. The scene at end of XMOW, I think that walking Xavier is just a mental image that he creates from inside of the airplane, since it maybe isn't so easy to get in or out of the plain with a wheelchair(especially if you'r just gonna ask the others to climb aboard). At the beginning of X3 I don't think Xavier is just a illusion. He might be wearing motorized leg braces, invented by Beast the super-genius, or he is wearing normal metallic leg braces that Magneto is making to move with his powers.
2)Double Emma Frost
The Emma Frost/White Queen familiar from the comics was seen in the FC. But there was another Emma with diamond-form seen in the XMOW, who how ever didn't seem to have any kind of telepathy, and she was also a sister to Silverfox(who happend to have telepathy-like mind-control-by-touch power). There is a theory that Kayla Silverfox and Emma were actually children of the Emma Frost from FC, But I have a better idea: There is a big telepathic battle(I like to think that it is the legendary battle between prof.Xavier and Shadow king) in which Emma Frost tryes to take a part, but she is getting backfire from the two more powerfull telepaths. She tries to protect herself by going into the diamond-form, but it doesn't help and she ends up being shatterd(like she did in end of Wolverine and the X-men tv-series). As she dies at blast, the last petty remains of her spirit are being attach to two little girls standin in the sidelines. As a result: the passive X-gene, that both girls have, activates, the other girl gets Emma's diamond-form and the other girl gets the telepathy(with a modification that it now requires a physical touch). These girls(sisters) would then grow up to be Kayla and Emma that we see in the X-men origins:Wolverine. This theory would be supported by the fact that Emma in the XMOW wasn't called Emma Frost for single time, in the ending credits she was "Emma/Kayla's sister", I know that she called herself Emma Frost in character-trailer, but trailers aren't really part of canon in my opinion.
3)&4)Xavier's and Magneto's first encounter/cerebro/Meeting Young Jean Grey
When Xavier and Magneto meet in FC they are both near 30 years old, in X-Men 1 Xavier said that he first met Magneto at the age of 17. This could mean that future movies reval that Xavier and Magneto did actually briefly met earlier but they didn't yet then make greater acquaintance. Other option is that Xavier just simply misspoke to Logan, said 17 when he meant to say 27.
In FC it was the Beast that built the first cerebro, but in X-Men 1 Xavier said that he built the cerebro with Magneto. This is easy to explain, since the cerebro that Beast made was destroyed by Riptide, all three(Beast, Xavier and Magneto) probably, during the manor training days in FC, planned and made blueprints in the manor, something like "In here we shall built our super-crebro".
In the end of FC Magneto and Xavier went their seperate ways and eventually became enemies, yet we see them together at beginnig of X3, both trying to convince the Young Jean Grey to join Xaviers school of gifted. This could be explaind by assuming that they were currently in truce, and when Jean's(class 5 mutant) powers were activated it was such big happening that both Xavier and Magneto felt that. They met, discussed and even Magneto had to admit that Xavier was the only one on earth who could teaches Jean how to control her powers stably, so they decied taht the girl would go with Xavier for now, and(to make the deal solid) Xavier asked the Magneto to come with him to meet Jean for the first time.
5)&6)Beast's second mutation/ Summers' family
Beast had his second mutation, aka the blue fur, in FC. Yet he was briefly seen on a tv-screen in X2 in his human form, and then again furry in the X3. Apparently it was originally mented that Beast's second mutation was caused by dark cerebro in X2. This could still be explained by a differnet guy. During his tv-interview in X2, the mutants werent so popular so it could be possible that Beast asked some one else to pose as him and tell his opinions to the public.
In comics Alex Summers was Scott Summers younger brother, but in movies this would be impossible, since Alex is in his 20s in FC and Scott is in his 30s in original trilogy that happens about 40 years later. So they aren't probably even brothers at all, now some have speculated that they might be father and son(stupid idea in my opinion). But I also heard rumors that they might be uncle and nephew, I think this would make a lot of sense, since they have same last name they are probably related somehow, and (according of the FC-backstories) Alex was in prison in FC because he had killed a man by using his powers. A man, who had tried to abducted him and his sister. Scott and Alex don't have a sister in comics, so could this sister be Scott's mother in the movie-universe.