Wild game is surprisingly better than factory farmed, relatively higher in Omega-3 fatty acids, lower in cholesterol as well as devoid of steroids, antibiotics and other additives because of their natural origin and active ways of life.
Pigeon and squirrel are very plentiful, very easy to shoot with a slingshot and quick and easy to prepare. They are generally thought of as pests so nobody will mind.
When you gut them the waste can be used as bait in fish traps which can be set at the low tide mark. At low water you can go and gather your catch then rebait wait for the tide to come in and go out again.
Pigeon and doves like all birds just need feathering and you leave the skin on.
With small game like rodents, rabbits and squirrels they just need gutting which is great bait for your fish traps. The skin, well, as long you roast off the fur and then scrape the last of it off the skin and fat has so much goodness in. But if you roast it up over the fire, and leave the fur on, chances are you'd be eating meat that tastes and smells like... burnt hair...so you must scrape all the burnt fur off but leave the skin and fat on which are the best parts. If you do skin the meat as you want to use the fur for something then you have to scrape the fat from the inside of the skin so add this to your meal, but better to leave the skin on and eat it all.
I've seen Australian natives roast a kangaroo by laying it directly on coals without skinning it, similar in Africa and other countries bushmen roast then scrape the fur off and eat the skin and fat.
For larger animals with thick, leathery skin, there is the toughness to consider. It's one thing to eat cow meat (beef) but it's something else trying to cook and eat cow skin (leather). It's tough to chew, has poor or no flavor, and slows down cooking/grilling time if you leave it on. This is not true with smaller game meat with thinner skin.
We recently cooked a whole wild pig - the skin was left on, but the small amount fur was roasted then scraped to be removed. The skin, once cooked properly, was mighty tasty.
So no, you don't have to deskin it. Just defur it. Same with dogs, foxes or similar as they do in many Asian countries.
Game shot in an unsuspecting moment is more tender than game that is chased and will also deteriorate less quickly. I now exclusively use trap for aquatic food and shoot land based food with my slingshot.
You ideally need a kill shot to the head. One second it's happy in its natural environment and the next second it's dead. Immediately after the kill, the animal must be bled. Work if possible in such a way that after removal of the guts and all waste parts you will merely have to wipe the cavity with a dry cloth. If internal bleeding has taken place, however, and fluids from internal organs have touched the flesh, scrape or cut the flesh as clean as possible and wash the areas water.