Unless the dog is posing an immediate danger you may not shoot it, this also applies to air driven weapons. I am not clear though whether edge of the property means outside or within, if outside then the 15 year old is breaking Oklahoma fire arms legislation, as he may only be in possession of an air driven rifle under the age of 17, on private land with owners permission.

The police need to handle this further, if all the witnesses cooperate then charges may be filed. In future you should secure your dog, the bar for what is considered aggressive enough behavior to be legally allowed to shoot a dog is very low. Your dog could simply run over to someone and that is more than enough, yes your dog is probably friendly, but the law is "reasonable belief you are in danger". So if a dog is running toward you, and you believe it is attacking, whether or not it is, then you may use lethal force. Dogs are property, they have virtually no rights outside of intentional abuse and neglect.

Of course you can sue his parents in civil court. Just because he was home alone, does not remove their liability. In all likelihood though, their home insurance will cover this.