Very hard to tell from the photo, but it looks to be either a travertine or a porcelain tile that mimics travertine. Removing a backsplash should be done carefully so you don't damage the countertop or the cabinets. I know some homeowners have tried painting backsplashes but I don't know how it would wear.

Whatever you do, you should change out the outlets so they blend with the backsplash--the white ones look out of place.

Remove it, and fix the drywall. THEN paint or replace it.

Yep - Remove with a hammer and chisel.