bigalc

To be honest, I don't think there is any one breed of dog that will be really good at all of the types of hunting you have listed. There are too many differences between what is needed of a good coon dog and what is needed of a good bird dog.
With that said, I think you may be able to reach some compromises. Here goes nothing:
I know of some coon dog breeds that also make decent squirrel dogs. However, it takes a special dog to learn to use his eyes on squirrels, and his nose on coons. Too many coon dogs will follow a squirrel up a tree with their nose, and never watch the squirrel leap from tree to tree. When you get to the tree Mr. Bushytail will be 100 yards away. I like blueticks for coons, but for a dual-purpose dog you might look atTreeing Walkers or Plotts. You might also look at curs or feists. I know they are well regarded as dual-purpose tree hounds.
Beagles make good rabbit dogs,and some have been trained as tracking dogs. But for a dual-purpose dog in this area I would suggest a Basset hound. Their noses are the most sensitive of all the dog breeds, with the exception of the Bloodhound. And from personal experience I can tell you that Bassetsmake great family pets. (In regards to the bloodhound, they are the best trackers, but they are too gentle-natured to fight a big boar coon. And they are too big to go after rabbits in briar patches.)
As far as finding a dog that would also retrieve doves, I think you would have to really work a lot with any of these breeds to get them to retrieve consistently.
One other breed you might want to think about is Jack Russell terriers. They make pretty good squirrel dogs, and they will really run rabbits (although mostly by sight). I bet they could be trained to retrieve pretty well too. However, I do not think they would make good coon dogs, and I do not know how well they would do at tracking.
As far as training, the best thing is to find some hunters in your area that hunt what you want to, and train your dog with theirs. To find these guys look for old pick-up trucks with dog boxes in them, and then look for a guy with no money nearby!
Best of luck!