Do you start with a character or do you start with a plot or a conflict or an idea?
A mix. I tend to be character-centric in that I try to figure out what my characters want and then don't give it to them, or have two characters that wants put them into conflict.
Have you ever created a character in one genre and rewritten it for another genre?
Oh, yes. I usually end up with several versions of characters trying to find the story that works best for them.