One of the walls in my laundry closet is open into the adjacent closet between the ceiling and the raw edge of the top of the drywall (about a foot gap). Is it safe to use over-the-door hooks to hang a shelf or shoe organizer from this drywall?
No. Drywall isn't meant to have exposed, unsupported edges. If you hang stuff over the edge, it'll probably fail - at the very least, you'll have drywall dust all over your closet.
Get some screws, and attach the organizer to the wall studs. (Since the top of the drywall is missing, you'll have no trouble finding the studs.) Take 5 extra minutes, and do the job properly.
And when you have a chance, slap the guy who built this travesty.