I just came across you post. My 2cents!
A dog does not have to be purebred to be a good hunting dog. It does have to have good bloodlines as far as the right mix. A beagle has been selectively bred for many many years to hunt rabbits. Same as a coon hound, and Lab. That will be hard to get a dog to do everything. The best all around dog is a Terrier. Airdale would be my choice. Terriers will retrieve from water, flush pheasants,hunt rabbits, tree squirrels, tree coons. I have trained springers, and I think if you started early at 6-8 weeks teaching to retrieve to hand, you could get them to fetch shot birds.