The easiest way is to have a partner and two drywall horses. Pin the sheet to the ceiling with your heads and nail the perimeter and best result is to have a follow up guy to screw it off as you and...
If the damaged area is small then a skrim or jointing tape can be used, if the damaged area is large then it must be cut back to the joists, replaced, jointing tape applied then skimmed.
You use an air compressor with an attachment to spray the compound on the wall.
Hog squash. You can apply it with a roller, however, it is easier and makes a better appearance if you use the air gun...