Wall Repairs

It sounds like you will have to cut out the entire damaged section. That wouldn't be a bad idea anyway as you want to make sure the studs are not damaged also. FWIW, many campers ago I tried to fix water damage behind a cabinet without removing the cabinet. I ended up removing the cabinet anyway. I was able to get by without removing the countertop. It made the repair easier since I had better access. As for trying to match it, I wouldn't try. Perhaps peel and stick tile or something similar along the entire area under the cabinets would work.

This is more serious. The wall board is wavy and not connected at the corner. I guess finding out if there is a stud somewhere along the wall might be my first step. Then cut out the bad piece but I have no idea how to replace it with something that looks ok.

Picture might help. Approach to minor cosmetic damage is different than fixing water damage which might have impacted underlying structure/insulation etc. Simple cosmetic damage can be fixed with paint/wallpaper etc - gets more complicated if damage is more severe.