Point of clarification: Are you buying the AR specifically for targets mentioned, or are they secondary? Nothing against the AR, but I'd suggest a .22-250 for those targets at that range. I have one, and it would fit your requirements perfectly. Check the ballistics. Flatter-shooting, little recoil, and delivers considerably more energy. Kind of half-way between the .223 and .243, and a significant improvement. I truly hope the AR is not your choice thinking that you will "walk your shots" to the target, which or course would be inhumane and sloppy marksmanship. And if those neighbor's dogs are coyote-size or larger, the .223 is marginal at the range you propose.
Muscles McGee makes a good legal point about the dogs. Be careful. I sympathize with your having livestock harassed like that; my lamb supplier had to start a whole new sheep flock because the old one was so harried by a "pick-up" pack of dogs that the sheep were too nervous to leave their enclosures to eat. She had to slaughter the entire flock. Plus the loss of those killed outright by the dogs. Quite a tragedy. Still, she was not (to date) killing the dogs. Don't know if she was trying to be a good neighbor, or whether the sheriff told her she couldn't kill them. This is also in Montana, where we have a fair amount of leeway in protecting our livestock. Good luck, whatever your choice.