I've always used nails so can't comment on staples. I will say I believe the idea of lifting the hide up on the nails is a waste of time. I never do that and never see problems. Think about this, When you case pelts such as otter, mink, raccoons and the like there is never any chance for air circulation on the fur side and they turn out fine.
I will say you should have the fur dry before stretching beaver or any other pelt. That said I have put pelts on the board when the fur was damp. The main problem with that is not spoilage. It causes the fur to mat and is difficult brush out after dry. Much easier to present a nice fur if the fur is dry before putting it on the board.
Just skinned out a wet beaver and it is ready to go on the board? Tack it on fur out with 6 or 8 nails and put it in front of the fan to dry the fur out. Just takes a short time.
On to the much discussed topic of over stretching. What is it? On beaver if you can get a nice oval stretch with no dishing in of the pelt it is fine. Often times you will see beaver that are "fishtailed" This is when the pelt is dished in on either side of the tail. The trapper was unable to pull the pelt out in this area because the tender hide in this area would tear. You can honestly say this was overstretched.
On the other hand I have heard people say you should have slack you can pick up in the middle of the hide when stretched. The only reason to do this is if you do not like money. You are cutting your own throat if you lose a size.
I listened to Greg Shroder doing fur handling demos for NAFA. He said grading on fur quality has more to do with length rather then thickness. If you have ever looked at a beaver that has been over stretched as opposed to one under stretched you would find it hard to notice much difference in fur thickness.
I am not advocating over stretching but if you can hold that oval line even though the pelt comes out snug it is not overstretched.