Well many of the issues arise from First Class starting out as a prequel, but when it was in development it became a reboot as the film makers decided to focus on ensuring they had at the very least a good film (the series really needed one after it had just had one dud followed by one incredibly bad film). Then after First Class was finished and they started developing the sequel and they decided to bring the old cast back as well (likely as first class whilst doing reasonable given the poor reputation the series had at that point was still far from the kind of box office needed to keep such a series going) thus making it no longer be the reboot it was intended to be.
Sorry but it's made pretty clear that First Class was always a prequel, not a reboot. Same mansion, same mystique make up, same Eric opening, Hugh Jackman cameo, etc.
They (the producers and the screenplayers) originally intended it to be a "soft reboot", that's why some things are very similar. I explained some differences on my site.
The X-Men timeline was altered 2 times, not just one (DOFP). FIRST CLASS was already a "divergence". Everything is explained here: