Letting a deer hang is perfectly fine under the right conditions. It ages the meat which allows naturally occurring enzymes to break down the tissue which tenderizes the meat and mellows the flavor. Ideal temperature for aging is approximately 34-42 degrees, if memory serves me right. At these temperatures, you can let a deer hang for 2-3 weeks. You should leave the skin on when aging as it prevents the meat from drying out. You can age at higher temps (into the 60's) but the aging process will be sped up and it will age in 2-3 days.
Once the deer freezes, the aging process will stop and there will be no further benefit from leaving it hang any longer. Also, younger deer will age faster because they are already more tender than an older deer.
I killed a young deer earlier this season and left it hang for 2 weeks before the temp dropped and the deer froze so I took it to the processor. It should be ready any day now and I can't wait to get it home and start eating! Should be really good.
If you want to get the full skinny on aging and processing, go to MyOutdoorTV.com and look for the game care section. There is a series of older videos on there that fully explain every aspect of field care to aging and processing and cooking venison. These are done by a Master Chef who specialized in wild game. It is really worth watching.
Hope this helps.
"It's not the catch or the kill that makes a day outdoors a success, but the opportunity to spend time with family and friends in God's creation that does so."