Like you, I have neighbors who can't (and won't) control their dogs. Their dogs are the only "visitors" that I have a problem with, and the owners know their dogs attack all kinds of livestock, yet still take their chances because they don't feel they have to contain them. Yes, they know I'll shoot them. I have no clue why they do this. But at least you have AC serving your area, we don't
When I called the sheriff, he said to shoot the dogs. Like you, it's not just our chickens at risk. I'd be one thing if I had to worry about natural predators, but knock on wood, I don't. I just have to worry about dogs at large from one neighbor. And I've seen these dogs harass chickens, cows, calves, goats, and rabbits. Thankfully, to date, they've only harassed my chickens. If they were harassing anything else, those dogs would have disappeared in some of many different ways.
I've seen some pictures of some awful damage that dogs have done to horses and cows. Writing about what they can do to goats and chickens is unnecessary, isn't it?
Even if your chickens are penned up, they're at risk of breaking their necks while flailing about in their enclosure when being harassed by roaming dogs. So even if you didn't free range your chickens, they're still at risk of injury even if the dogs can't actually get in with them.
Since your neighbor knows darned well that his dogs are harassing your horses, I wonder if you might consider a livestock guardian (or two). Are you able to contain them in your horse's pasture? Remember that each one will cost you monthly meds, like your house dogs, but if it's worth it to you, they are great to have around.
But the part about the dog working so hard to get out really concerns me for your family. That dog is really a hazard. I just hope someone does something about that dog before he gets to a child. Once is too late IMOHO.
I do commend your neighbors for trying to keep the dog contained, but I'm sorry, they need to work harder. It's their responsibility to contain their dog. He's a dog for heaven's sake. Do they mean to say that their dog is smarter than they are? If they can't contain the dog, it needs to get gone. IMOHO, it needs to be put down, not just made to be a problem for someone else (and their neighbors). A dog that can't be controlled is just plain dangerous.
Good luck, and stay safe!