Paint acts as primer.....just paint away....or prime first. If you paint you will need 2 coats unless you use a primer.

Umm....The paint is peeling off. Primer acts as a glue.

Feather out the edges where the paint has peeled. Use a bit of spackle and sand and feather that in. Once done, fill in your nail holes, or anything else you need filled in, and make sure walls are clean and vacuumed well. The put a good coat of quality primer on the walls, and repaint. No, you do not have to peel off all the old paint. Paint does act as a primer, however, by using paint as primer, you will need to use many more coats of paint, so it costs a lot more to do that. Primer is cheaper and it seals, so you get by with 2 coats of paint. What you have is not a huge problem, and very simple to repair.

Should I prime the sheetrock and keep painting or do I need to peel off all the paint first?

Absolutely do not peel off any more paint/paper.

If I'm following the story, no you can't just paint over it.

You're going to have to skim coat where the paper peeled off in order to even out the wall or else you're always going to see a line of demarcation between what is now essentially two different substrates - the part of the wall with paper and the part of the wall without.

Paint is not primer.

And primer does not act as a glue.

I would hire a professional painter to fix this because there is an art to skim coating and repairing dry wall.

If you want to DIY, take several pictures and go to a professional paint store - not a home store - and ask them to recommend process and products.