Most rural jurisdictions have laws that allow a dog that is harassing stock to be killed. Do those laws hold in the rural part of your county? If not, do they at least hold in other counties in your state? Does your jurisdiction have a leash law? Was Gapp in violation?
It does not sound like you have been keeping your chickens in violation of your local codes (correct me if I'm wrong here) ... so is your police chief suggesting that your chickens constitute an attractive nuisance because they may attract the attention of dogs that are illegaly running at large? That does not sound like an argument that will get him far, as the same could be said for letting your children play in their yard or ride a bike down the street. The fault here was with the dog owner. I would ask the police chief if this is the case. Ask him if he considers the chickens an attractive nusance because they attract the attention of dogs running at large. If he says "yes", ask him if you get rid of the chickens, must you still fence in your children because they are also an attractive nuisance to dogs running at large? He'll suddenly defer to the city attorney, who will know wha the heck an attractive nuisance is, and he'll find himself straightened out on the issue.
I think there's a bit of placing dogs someplace between chickens and people in terms of value and rights going on with your officials. However, unless there is an explicit law to that effect (highly doubtful), dogs are animals in the eyes of the law. So are chickens. They are both property.
As for Gapp getting a lawyer and pursuing civil action, I'm not surprised. If he can get any sort of result, whether it be a judgement or a settlement, he can tell himself it was your fault and not his that his dog ended up dead. The burden of proof is on him regarding his claim that you "hunted" his dog. Your having called the police yourself immediately after the incident, and before Gapp arrived on the scene would cast substatial doubts upon his allegation with any reasonable person. I am not a lawyer, but I can't see him being able to sufficiently substantiate such an accusation to get a judge to agree to a trial. Has Gapp's lawyer told you that they intend to file suit, or is he merely hinting at it and sounding threatening? That might actually be pushing things close to being actionable on your part. Between the false accusation and the mailbox stuffing, he could be getting himself in trouble.
I don't honestly see what he can go after you for. The cost of replacing the dog at most. He's going to be hard pressed to make any believable case for "emotional distress" over a dog he had for about a week, that he cared enough about to let it off leash and out of sight for what, an hour or thereabouts?
Can you post email addresses for the officials you referenced? I'm sure many emails will be sent if you do.