backspalsh

I am replacing my backsplash with 3x6 travertine limestone tiles. The removal of the old tiles has resulted in damaged drywall. Some of the actual drywall paper is peeling off and there are some gouges as well. My questions are do I have to remove and replace these sections of drywall or is there something I can put over it before tiling? Also is white thinset better than mastic for installing this kind of tile?

You don't want to use mastic on stone tile, use thinset. Go get a six inch broad-knife and gallon of joint compound and fix the walls. Skim them with joint compound using the six inch knife. That will lock the paper fuzz and fill the divots at the same time.

Once that has dried completely then again skim the walls with a flat trowel and thinset. Let that dry and go to tiling.

Is this okay if the drywall is heavily damaged?
The drywall seams to have spit open in some places.
The damage goes beyond just the drywall paper.
The old tile just didn't want to come off and tiling over it wasn't an option.

FINE! Take out all the damaged drywall and reinstall new drywall. Be sure to cut the old on joist centers and tape and bed the new after it is installed. Cut it neatly below any cabinets and above countertops. When cutting into the old drywall be sure not to cut any wires or cables that may be inside the wall cavities. There could also be plumbing improvements in the walls so use caution there also.