Since you already started down this long road in a rental, you have to keep moving forward, because if you didn't, I would say don't touch a thing. Since the wall now needs to be repaired, I would either replace all the dry wall or skim coat over it. Prior to skim coating, you need to remove as much loose pieces as you can, then seal it with something like Gardz, skim coat, then seal, prime with shellac on the damaged areas, then prime the whole room.

If you've never done it before, I would just hire a handy man to replace the dry wall.

Sounds like you're dealing with old wallpaper on top of even older plaster. Hopefully a pro will chime in here to advise you on your options for repair. One option you might want to explore is anaglypta, which is a patterned wall covering that you can paint over. I've never used it, so can't comment on how well it works, but I've heard about it for years. Good luck on your project.

The crumbling wall needs to be repaired with plaster, not a skim coat of joint compound. It's not difficult, but should be done is a few layers (and roughen the under layers so that the next layer has something to grip on). Your final layer will take some care, and then there's a longish wait for it all to dry thoroughly so that the paint adheres. Even plaster that seems dry may not be and the paint will bubble after a few days.

It might--might--be easier to cut out a clean area down to the lath and patch with wallboard.