I try never to skin a beaver, the day I catch it.If you lay it on its back in a cool place,the blood will coagulate in the head---thus no bloody mess.
I never hang them, and I never trough them.I use the loppers on feet and tail--I find the tail gets in my way.
I make a mid line cut from the tail area to the chin,making sure I do not cut to deeply and split my castors.
I then start skinning along one side,I do not stop half way---I keep going all the the way across the back--when I finally start on the other side,I am already 3/4 done.
While I start with the beaver on its back,I don't keep it there.As soon as I can,I place it on its side.I let the weight of the beaver keep constant pressure,on the pelt(it wants to roll over--but I am holding the edge I have already skinned)This makes for a much faster and easier job.
The best advice I can give--develope your own system--what ever works well for you is best.
Just remember--the first thousand or two are the hardest---after that, its a piece of cake.
I would rather skin a beaver ,than any other animal--and that includes muskrats.
Tom