Taking things in reverse order:
With the large areas firstly remove any remaining loose plaster. Trying to patch without doing this is a waste of time. Then with the large areas you'll need to replaster. You can either take this as an opportunity to learn or pay someone. I won't tell you how to plaster as it's really a skill you have to be shown or practise yourself, though I do have a couple of bits of advice. Don't try to plaster the whole wall in one go. Take it in stages and make sure you dampen the wall you're plastering onto as it helps the plaster stick.
With the areas missing the top coat of plaster you will need to reskim. This should be done after fixing the base coat. You might find a plasterer willing to skim over your base coat - discuss it before you start. This will give you a good finish but should cost less than hiring a plasterer to do the whole job. Getting a good skim finish takes a lot of practice.
Finally the areas with only minor cracks and bumps should be filled and sanded before proceeding.
If you manage to get a good surface then painting is the best option. If not then paper the wall with two layers of lining paper. The first runs horizontal and the second vertical. This will smooth out most of the remaining unevenness in the wall. Before papering wash the walls with a weak glue solution this will stop all the glue from the paper being absorbed into the plaster and the paper lifting off the wall.
I should have added that you need to leave the plaster to fully dry out before painting or papering. That might have been one of the reasons why the previous repair has failed.
UPDATE
The quote you have doesn't seem unreasonable but without knowing the exact area you need replastering or your location it's difficult to say for certain. However, we shouldn't be getting into commenting on exact quotes as they are always going to be too localised (in time and space). Get another quote to compare it against - that's always going to be the best option. Also as I said above see if you can get the plasterer just to do the final skim - it should come out cheaper.
First, make sure you get all (and I do mean all) of the loose paint off the wall. If you get a fairly flexible wallpaper scraper you should be able to bring most of it up. Keep going for that extra 5 minutes after you think you've finished.
Then sand the remaining paint to feather the edges so that they are as smooth as possible.
Next get some filler (spackle I think it's called in the states) and fill in any cracks and large holes. You can also use it over any edges that still seem harsh. When it's dry a light sand will smooth out the imperfections - but don't let it get too dry as it can go very hard.
Then wash the walls down with warm (mild) soapy water, then rinse off with fresh clean water. Allow to dry then prime the wall with either a specialist primer or a watered down coat of emulsion. This will also help find any last imperfections which you can fill and reprime.
Then the wall is ready to paint.
Best Answer
If you're using an electric sander then you want to keep that moving all of the time & not linger in any areas. Lingering will heat up the paint & cause it to release. If the paint is peeling off in strips you may also have a poor paint job where Priming before Painting wasn't practiced. But, you won't make that mistake.
You can hand or pole sand with the sandpaper or sanding screen to avoid so much heat & it will also demonstrate how much heat sanding creates. Additionally, you can Wet-Sand with nothing but a Wet household Sponge. This doesn't create heat or even dust. But, definitely remove any loose paint edges as best you can.
If the paint keeps pulling off where you haven't sanded & you have plenty of Joint Compound or Spackle to feather the edge where you finally find sound paint, then rip off as much as possible...This part assumes you have plenty of Primer & Paint to tackle the larger area(s), once with Primer & twice or with 2-coats of Paint.