I just finished reading this entire thread, what a complex problem. I wouldn't dream of allowing my dogs to bark all day, my music to even be heard by the neighbors, or having a rooster waking them up every morning. Many of us have different sleep patterns, and to us 5:30 is the middle of the night. I'm not saying that you should get rid of your rooster at all, but it's very difficult to get from your post what the situation is as far as proximity to each other or what the "sparring" entailed. There are many reasons that someone would carry a paper bag, I've been known to myself to pick up any trash I see on my walks. That he would have it to haul around a newly dead rooster seems a bit far-fetched. Also, if you were bickering he may well have just been looking at your house afterwards like "can you believe that?" and processing what was said. We can't know what's going on in someone's head. I am sure you felt threatened, and perhaps he was intimidating you on purpose, but perhaps he wasn't either. I agree that you took the right precautions here, but you may need to also deal with the fact that your rooster might be a nuisance to one of your neighbors and is driving him nuts. What you do with that is up to you entirely. I'm not judging, just giving my 2 cents.
For me, just because we get college kids with loud music rolling through the neighborhood, or dogs barking at times, it doesn't mean that I can do whatever I want without showing any courtesy. I much prefer a neighborhood that looks out for each other and talks to each other over the fence, and having that happen is as much up to me as anyone else.