You CAN install tile over drywall.
TCA (Tile Council of America) approves of installation of wall tile over gypsum board.
You don't necessarily HAVE to use a backerboard system.
I would NOT use a mastic or pre-mix, but rather a modified thinset.
You are going to get technical for the top row over that arched niche over the fireplace. Gravity is a mother on wall applications.
Good luck on your project.
Cupan Custom Tile and Paint