A smaller project costs more per foot than a larger one, plus you probably have a change order fee incorporated into it if that wasn't what was planned from the beginning. My good tile guy charges around $15 a foot for a simple backsplash. Even with the change order fee of maybe $200, plus the GC's fee on top, your quote seems out of line for ceramic. Glass would be more, but not even that much!

600.00 might be a bit high, but not unrealistic. Keep in mind a couple of things. First, it's tougher doing a backsplash because you've got to bend over for the whole thing, but at the same time lift your head up, as well, so by the time it's done, both your lower back and the back of your neck are CRYIN! Secondly, in most backsplashes there are more cut tiles than full, due to the small area, and the fact that every foot or two, there are outlets, switches, or windows, and all the cuts have to be spot on, because they're much closer to line of sight, so they'll be more noticeable if they're NOT perfect. Lastly, most installers have what's called a "daily minimum". That is, a minimum amount they need to make for a day's work. For some small projects, it's just not worth it. But for a backsplash, like I said, it's not an unrealistic price. (I will say this, though. Unless there's a detail you're not telling us, like maybe it's tumbled travertine subway tiles, or something like that-- my price would be about 150.00 less than his)

Is that just for installation or materials too? We had 22sf of backsplash tile installed two years ago in high-priced San Francisco - we got three bids for installation, the highest of which was around $2000, and the lowest of which was around $1000. In this case, we went with the lowest, becuase they got great reviews, and they did a great job. That price didn't include the tile, which was $40 psf Fireclay crackle tile.

I'm not a professional and it depends on the size of the area to be repaired, but it's pretty easy to cut out damaged dry wall and then mud the patch seams. You want to do whatever will maximize the clean, level surface. And, yes, you will need to feather and sand your mud until it is dead level. I wouldn't paint the mudded, repaired drywall before tiling, at least, I never have. And I've never primed drywall before mudding it. If you've only read or watched videos, be sure you practice. If your repair area is large, even if just 'paper', maybe hire somebody?