I don't have proof, but I do like all my venison aged at least a few days, a week is even better. Not sure if it improves the flavor, but I do think it makes it a bit more tender, but that is very hard to quantify since you would have to compare the exact type of animal; aged vs. unaged to know for sure.

A rank old buck doesn't improve much from aging. However a nice young cow elk, can taste almost like beef when taken care of properly and aged. Same for antelope. We have meat that is just about fork tender. 

You would have to ask a butcher for the actual temperature for aging meat but I believe that it is 41 degrees or slightly below.  What you want is it low enough to allow the meat to start to decompose but higher than freezing.  The longer that you age the meat usually the more it will break down and be a lot more tender.  But the key here is a constant temperature to do it and most home butchers don't have the cooler to do it.  The only elk that I have had a processing shop butcher was aged for 14 days before they started to cut it up and the meat was fantastic. 

My dad use to hang a deer with the hide on and not butcher it until there was mold growing on the inside of the ribs and I never remember a bad piece of meat off of the deer.  He would also do the same to a sheep that he would buy from a local rancher.  He would also wrap the animal up in blankets during the day to keep the meat cool and unwrap them at night to let the cool air get to the meat.   

Like the others have said it is very difficult to tell with the many variables we deal with. If it's cold enough I will let it hang for as much as week before I cut it up but if it's too hot out as it usually is in the earlier seasons then we will be cutting the next day. But again the variables of how old the animal is, what knid of animal, and a big one in some areas is what it's been eating or running have a bigger effect on the meat than aging can change.

The absolute strongest worst tasting deer I ever shot was a spike that had to have been eating something not in a normal diet. The most tender meat I ever had was one quarter that  actually hung for a couple of months but was frozen solid for a good part of that time. When it started to thaw it started to mold on the outside as I stupidly covered it with plastic. I cut it up and it was far more tender than the rest of the deer.