I have spoken with a lot of different Deer hunters in my area and everyone that seems to know anything about cooking says to hang the deer.
If it is a young deer or doe it should hang for about a week in Temps between 34-38 degrees. If is is a big Buck then two weeks is the maximum.
I myself haven't tried to hang deer because I didn't have the set up until now. I will be skinning it, and quartering it and keeping it in a cooler above some ice (make sure it stays dry) and let the water/blood drain out. Replace the ice as needed to keep the temps low.
Don't freeze the meat right after you shoot it because the Rigor has not left the body yet. If you ever had tough venison after you got it back from the processor they usually freeze them right away.
Keep it cool and dry. Let it age and enjoy it.
Of course you don't do this with the Backstraps because they are more prone to dry out with aging. I take mine out right away and put them in a ziploc bag and wrap them in butcher paper and put them in the freezer. Leave them whole and cut them as needed.
Hope this helps!
Oh, and the sausage looks great! I will be making some here soon.