I want to share my story of a similar nature..................... 1990
I AM the one who DID shoot the neighbors dog. Of course that was after 12 complaints, at least another 12 warnings and then I had, had it! Not only had this dog killed several of my chickens, ducks and almost took down a new foal. The police wouldn't do anything to help me.
So I shot the dam dog. I was told I had killed him. Of course I was a bit confused because it wasn't a kill shot (yes I was that good) I hit him in the hind quarters. An eviction notice later (landlord and owner of mutt were best friends) a 500.00 vet bill and the dog came back to life and was roaming in his yard.
I refused to pay the bill, fought with the landlord to the point of getting ready to shoot him in the hind quarters too! Had to move and to add to the whole frustration had my guns taken by the Sheriffs dept. You guessed it. I was ready to shoot all of them at this point.
This is one of those situations that has no win to the whole thing. Your going to have to file the restraining order and let them know about the threats to you and your child. Get their guns removed (the threat alone will make that happen). File a small claims action for the loss of your animals and demand that they either tie the dog up. or find him a new home. He is not a farm dog. He is in fact a danger to everyone's stock.
I really like the paintball idea, BUT I have a feeling the repercussion will be worse. The laws have changed a lot. A persons personal property has higher value to it now than it used to. Back then, all I heard was "It's just a chicken" Or "he hardly made a mark on that foal" I have a bit higher hopes that the law now will force your neighbors hand in the whole thing, and I would imagine they might have to get rid of the dog.
OH one more thing.... Try to catch the little beast and call animal control. They will come out and remove the dog and document the incident. That will get it to a judge real fast!
Good luck, keep us informed.