He kinda reminds me of various ranchers I know.
Likely he thinks you're the a-hole, these various interactions for any reasonable person are totally benign... any reasonable person wouldn't even notice.
He's clearly not reasonable, and given the above list I'm assuming sees his property as sacrosanct.
In these situations being in the right doesn't matter. I mean it might matter in a court of law, might matter to you, but in the context of navigating this relationship you have to realize that he thinks he's right, and likely to an absurd degree.
My conjecture, and obviously arm chairing at a distance, the most frustrating and yet probably most effective way to deal with the issue is just to capitulate.
He wants an apology for all the slights, anything you say to try explain mitigating circumstances will likely just piss him off. I think in large part folks miss the point of an apology, it's not to 'atone' for what you think you did wrong, the point is to make the aggrieved party feel better/heard.
"I'm sorry my dog chased your chickens, clearly I was negligent in not keeping him restrained. I will talk to my kids about being more considerate of others in public... yada yada..."
Clearly the dude is a huge dick/ piece of work, but I imagine if you apologize for all the things he's frustrated about, regardless of how you feel about them, you might be able to de-escalate the problem.
I think that's the route you'd have to go to bury the hatchet.
Up to you whether that is a bridge to far, and it legitimately might be... me I would just put myself in the mind set that I'm trying to talk down a toddler and just try to put an end to it. Folks can only get to you it you give them the right.
...
but yeah all that being said #*^@#* that guy