I May Have Missed Something...
Spoilers ahead!!!
In the post credit scene, Charles is alive... how is this? And who is Moira?
As I said, I was interrupted part way through and missed a bit, so thanks in advance!! :-)
Charles is alive in the main bulk of the film. Until that moment.
The pre-credits scene was supposed to be set in the past, hence his walking.
Moira is a doctor. She was played, briefly, by Olivia Williams in X3: The Last Stand. And in X4: First Class Rose Byrne plays her.
Xavier wasn't walking in the mid-credit scene. All he did was turn his head while remaining horizontal in the bed and say; "Hello, Moira." It's also not set in the past; the idea is that he saved himself by transferring his consciousness into the brain dead coma patient. They set that up by talking about the ethics of doing that very thing in class during the beginning of the film.
You seem to be confusing the mid-credit scene with the opening scene where a digitally de-aged Magneto and Xavier walk into a young Jean Grey's house and seek to recruit her for their school.