Peeling some old paint off - as in the paper barrier on the sheetrock has completely came off? Or it just looks bad from where all the glue was removed (which was probably the biggest pain in the ass?) Im sure there is someone here that knows better than I do, but I would start with killz to make sure the glue residue isnt funking anything up, then sand it with a wall sander, preferably one with a vacuum attachment, then skim coat it as you're saying, sand it again with that same tool (which you can rent one if you're really interested) and then paint it up nice and pretty. Again, Im not sheetrock guy - just enjoy getting my hands dirty on my own terms.
My first inclination would be to find out why there is so much moisture in the HVAC. Cleaning it up is one thing, but your evaporator should be sapping out that kind of moisture, not spreading it to the house. This time of year with the rain we've had in VA - my systems are draining an obnoxious amount.
As for the black mold - it would be worth it to test and see if thats really what it is, Im sure. Ive caught it early enough in older properties that it never developed into anything more than a quick cleaning.
But again, Im not an HVAC or mold person.