Am I the dick here?
Yes. You fucking shot someone's cat and then lamented that you "didn't kill it". I don't like cats, but you're still an asshole A) for doing that and B) for saying that.
I understand you may be concerned particularly about the dog but it sounds like it was probably an honest mistake.
Let me explain why I think calling the cops on your neighbors is rarely a good idea unless there is actual crime against person/property in progress, or someone is in imminent danger...
Calling the cops on the people who live next door to (or down the street, or around the corner from) you generally doesn't do much to promote positive relationships/community especially when it is over relatively mundane/trivial shit like "someone's cat pissed on my car" or "someone left their garage door open for more than 2 hours in violation of Subdivision Bylaw 3634 section 15b." When this happens you quickly become "that asshole who called the cops on us when our puppy escaped out the gate". And you have to live next to them for as long as you both live there.
It's a much better use of your time to try and resolve things man to man, than to call the cops to intervene in every piddly little neighborhood dispute.
You've sent them four or five emails but how many times have you talked
to them personally about this? How many times have you let them know what your concerns are? How many times have you civilly
told them something like, "Look, I'm not trying to cause any trouble here but I've asked you a few times and if you don't keep your pets in check you're not leaving me any other options than to call animal control"?
Now, if you've done all these things and these people are either idiots or assholes and they simply don't respect you or anyone else in the community enough to take care of their pets and follow basic rules of human decency, they've left you no choice but to call the cops.
But you're still an asshole for shooting their cat.