I would handle the introduction of mutants in the MCU similar to how the X-Men: Evolution cartoon did back in the early 00's: mutants have been around the whole time and have largely managed to remain hidden, but now due to a public incident of some sort, have been revealed and the public is freaking out. I don't think it is a stretch to say that the X-Men have been around in some form, but as they were trying to keep mutants hidden, have stayed out of the public eye.
My ideal team line-up would be the original five plus Wolverine, Storm, Nightcrawler, and Colossus. Granted, I'm not sure if that would be too big of a line-up for the first film. In which case, I would probably remove Angel and Colossus for the sake of condensing.
I also wouldn't include Magneto, the Brotherhood, or Mystique, but would reserve them for sequels. I like the idea of starting off with Mr. Sinister. He would lend himself to providing exposition on the history of mutants, I could see him being responsible for the incident that exposes mutants to the world, and his dynamic with the Summers would help put the spotlight on Cyclops.