I have 6 inches of moulding trim around the ceiling. The ceiling has

If the builder did the work, the piece is nailed (like the floor molding), with headless nails. You're going to have to pull it off if you want to replace the drywall. Use a small crowbar and loosen it first. If it isn't more than 10 feet, it should come off in a single piece. Get some help to catch it as it drops. Fix your wall and then put it back. You can either re-nail it or use something like liquid nails to hold it in place. Might also need a new coat of paint.

Hi M, Here's a how to video. They are removing floor trim, but the same principles apply. https://www.youtube.com/watch?v=6p6Xn06AfqY

Good luck!

A photo would help. If the drywall is not soft and just stained you can paint over the stains with a stain blocking primer like Zinsser 123 or BIN to cover the stains. If the drywall is damaged and does need to be replaced best to try to carefully remove the molding. Also depends what the molding is made off. If its wood it more likely is nailed. Another option if you don't want to remove the molding you can very carefully cut the drywall out where the molding meets the ceiling with with a razor box knife or drywall saw. Replace the drywall and caulk the seams where it meets the molding.

Ditto . . . No way to fix ceiling without removing.

Just a thought, but if priming and painting will not work could you leave the perimeter of the ceiling alone and just come in about 10 to 12 inches from each wall and add thin (.25 in) drywall sheets to the center to cover stains and trim it out with molding. Much of the outcome would depend on your skill level.

Is the ceiling textured or smooth? If it has a knockdown texture, you could probably prime and paint, then get a can of spray texture in the right texture like a knockdown.

HAHA! Tell the boy it's not nice to flush a superhero's cape down the toilet. Batman's is gonna get him!

Well I would try this first, you can always cut it out if you don't like the results. Another technique I've used if the surface needs smoothing is to take modge podge and sheets of very thin white rice paper, and decoupage a patch over the area after removing any texture first. Then prime and paint, and use your texture.

I would think an art store or craft store. Seems like Amazon and Etsy have it... https://www.google.com/search?client=opera&q=white+rice+paper+for+decoupage&sourceid=opera&ie=UTF-8&oe=UTF-8