If you have plans of your property it usually has a T pointing into your garden on the boundary. Not familiar with the Party wall Act, but reading the extract you have supplied suggests the the fence is shared responsibilty, even if entirely on your land. I take it you have been informed by other neighbours that the children are tormenting the dog. If it was my dog I would certainly point out to my neighbours that the fence wouldn't get in a broken state if they controlled their children and they were not allowed to torment it. You state that the children are also banging on the fence , surely they are doing equal damage. Is there a reason why the dog is being left out, does it cry and bark, or become destructive when left in the house, if not, then cant you keep it in , it would be kinder , rather than leave it outside to be wound up. Especially if it will be only inside for a few hours and not all day. Our dog is perfectly happy left inside, providing she has had a good walk beforehand.