The nominee ALWAYS has the right to remove the chairman, and many do. They don’t whine and brag about it, though.
Nope. The chairman can only be removed by a 2/3 vote of the RNC. Now in most years, the nominee would have a lot of support from the members of the RNC, so they would go along with his desires. But Trump has made sure that he has almost no support from the RNC, so they have absolutely no reason to go along with him on this.