Yes, hang it right side out on the hanger.

Actually what I do after diving and rinsing off my suit is to hang it upside down by the feet as opened as I can get it. I have a little portable fan I use to blow some air up inside it. When it is completely dry I hang it right side up and leave it that way if I am going to use it again anytime soon. If I am not, I will clean and lube the zipper, talc the seals and put it in the bag. As far as putting it in the bag, just drop it in feet first. I try to place the seals in a safe place far away from the zipper or inflator.

As far as zipper open, or closed, I have heard people tell me both ways. I think I leave my dry suit (front zip) unzipped, and I leave my wife's (rear across back zip) zipped. My zipper seems much more flexable than hers does.