You definitely should use more than one coat of joint compound. I like the All Purpose regular type, but this shrinks a little more, but is also more forgiving (heaps up less). Apply to gouged out areas, skim off with a wide enough blade taping knife (4", 6" , or 12"), let dry, scrape off any ridges sticking up, and repeat with another coat. The mud will shrink in after each coat. I find that with regular all purpose, 3 coats are needed when filling holes. It does not take long to fill this way, just some patience to wait for the coats to dry.
You can leave the mirror up if you can finish up to the edge of it - it makes it a little more difficult to finish, but it can be done. If the mirror extends over the surface and does not allow you to finish properly to the edge, you should remove it.
With respect to tile, assuming the area is not going to be getting a lot of water, simply finishing and priming should be sufficient. In water areas, like a shower, durock cement board is needed under the tile. In your bathroom, a mold resistant drywall should be used and also use mold-resistant primer and paint - Zinser makes some I have had good luck with. In our bathroom I actually used Durock in place of my drywall right next to our tub where kids splash water - I seamlessly taped and finished this to merge it with the regular drywall and it turned out great, and holds up great to water - you might consider doing this as well right next to your tub if water is often splashed.
For some tips on finishing drywall, if you need them, see Drywallinfo.com.