Can I hang over-the-door organizers from exposed 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.