Check out your local ordinances as to what you would legally have the right to do. If it would be legal to shoot a livestock-killing dog, let the neighbors know this. If there are puppies in their household, they may be breeding this dog, and not want to lose that valuable commodity. If they do not take that seriously, itemize everything out carefully, and gather whatever evidence you can, and pursue them in court. Let them know that it is more expensive to raise a bird, plus the emotional involvement, than they may realize. Let them know you are serious, and that you will pursue the matter.
As for paintball guns, if you really feel you can get away with it without them knowing for sure that you were the one that did it, then that may be a serious plan. If you hit their dog, and they can prove (somehow) that it was you that did it, you may run afoul of animal abuse enforcement and be fined.
Strong security and loss prevention in your yard is of primary importance.