I hang my deer by the antlers or head, because that is the way my father taught me, forty years ago. Also, I think a deer just looks better with its head high and for me it imparts more respect for this beautiful animal. Like not cutting the head off a beautiful Brook Trout, which is a fishermans way of showing respect for this sporting fish.
I process all my own meat and I am convinced that the idea about blood draining into the hind quarters is a lot of balogne. Besides hanging head lets you wash down the inside of the deer with cold water, once it is hung, and the water drains out the hole let by removal of the anus.
As far as skinning, I have never skinned a deer head down, so I do not know if it is better or not. But I do know that the most hair will get on the meat at the location that you start you cut and by skinning down toward the hind quarters, I get no excess hair on the bulk of the meat.
By the way don't forget to remove the tender loins, ASAP after hanging it (regardless of which way), so they do not dry out. The best way is to use your fingers to work the tender loin away from the back bone starting from the top (hung heads up) and only use a knife to cut this choice meat free at the bottom.