Hanging times

Hanging meat is not an exact science.  Much of it relies on the ambient temperature and personal preference.  When I first started with pheasants I was always concerned about the smell thinking that the longer it was hung the smellier it would be when I gutted it.  As a result, I didn't hang them but this didn't mean that they didn't always smell.  Sometimes they did and sometimes they didn't.  If you are sensitive to the smell then the best thing to do is to gut them outside.  Once you have done a few birds you will get used to it and, provided the bird hasn't 'gone off' you will even come to associate the smell with food.  Let's face it, you probably don't turn your nose up at the smell of raw beef or lamb.  Each meat has its own smell, just as most cooks can distinguish between the smell of beef and lamb, I can distinguish between the smell of pheasant, rabbit and venison.  Admittedly, the entrails are generally a stronger version of this but, like I said, you do get used to it.
So, going back, I didn't originally hang pheasants believing I was making life easier for myself.  I was wrong.  Hanging is important for a couple of reasons.  First, it enables the feathers to be plucked easier.  Pheasants are a pain to pluck if not hung.  The skin is very friable so tears easily and most of the feathers seem to have an iron grip.  Pigeons, on the other hand, almost drop their feathers as you look at them.  If you want to spare your fingers, you would be advised to hang your pheasants.
Second, hanging game helps to tenderise it.  You probably won't know how old your animal is once fully grown.  There is a difference between eating a young animal and one that has been around the block a bit.  The former will be tender without any hanging but the latter may need a good set of teeth.  Hanging can relax the muscle fibres and solve this problem.
Third, it can intensify the flavour.  We have all heard about meat being hung to develop the 'gamey' flavour but you don't need to go that far if you are not keen on your game being too strong.  A beef steak cut from an animal only hung a couple of days will taste very different from the same steak hung for 21 days.
Last, with larger mammals, hanging helps to draw out the moisture in the meat so to go back to my steak analogy, if you like your steak rare or medium rare and you cut through one that has only been hung a few days it will bleed all over the plate.  conversely, you can cut through a steak that has been hung 21 days and it won't bleed at all even if it is 'bleu'.
OK, I have possibly convinced you that hanging is a good idea.  Now you need to know how long.  this is a much harder topic to discuss because, as I said, it will depend on the ambient temperature.  Try to hang your meat in the coolest part of your house or a shed if the weather is reliably cold during the winter.  Remember that sheds are liable to extreme changes in temperature as even in the depth of winter, if the sun comes out, they can get quite hot inside.  You can use a greenhouse thermometer if you are concerned.  Ideally, you want the temperature to remain below 10ºc. If it goes above that you might need to shorten you hanging time.
As a general rule of thumb, I hang pheasants, partridge & pigeon for between 1 & 2 weeks.  Duck only a couple of days.  Rabbit can be hung a couple of days but I generally deal with them straight away.  If you get a goose, you must eviscerate it first as its size means it will take too long to cool down and so the stomach can fill with gas and explode if you don't.  Venison can take a lot of hanging but, like geese, you must eviscerate it first.  If you have the luxury of a large cool hanging space then you can hang it for up to a month.  I don't, although a farmer friend of mind has now kindly offered her cool store for my next one.  As it was though, my venison, up to now, hasn't been hung at all but I have been lucky in that both the deer I have had were young so very tender, but the meat was quite bloody as a result.  Any game that has been eviscerated, or any that have open wounds, will need to be protected from flies especially if not during the winter.  Ideally, if the skin is broken and exposing large areas of muscle, it is a good idea to use a needle and thread to sew the edges together to stop the flesh drying out.  This is particularly worthwhile on pheasants.
I like the game to have lost all of its stiffness gained from rigour mortice at the very least and the warmer the temperature the sooner this will happen but basically, it's a matter of trial and error.  Building up the confidence to let the meat hang will come with time.