I read through the thread again. Really sharing your frustration and anger. I posted early on in this thread that you needed to secure your fence and fix every hole, and if the dog gets through then it is property damage. I know its not easy, but why don't you have a fence that offers privacy and security? Is it too expensive?
So then, now you have a new coop. Maybe you can fence that off? The 300' minimum is to the neighbors dwelling, not to the property line. That means they should be measuring from your coop to his house for the setback, not to any property line.
You said the inspector is from the Town, but you live in the incorporated village, in fact the oldest village in the county. You need to do a history lesson on where you live and what rights your village board has. I know a town inspector should not have any jurisdiction in your village. I also don't think any village board has the power to be doing what they are doing to you. The village board gets power to regulate certain dwelling aspects from state statutes, and the state statutes for new york do not grant them the right to do what they are doing. It's almost as if they are trying to apply the town ordinances to you, but you live in a village. Their governance does not cover what animals to raise in your property. It simply is not legal for them to have meetings passing whatever rules they like to apply to your private property.
A lawyer should send a cease and desist letter to all of these people. They are harassing you.
Now back to my frustration with you is that why can't you get your fence built so that the neighbor can't even see in your property and his dog can't enter. Build a 6 foot or higher wall between you and keep everyone else off your private property. I think your neighbor would probably be silent if there was a fence, that is if the roosters aren't a problem now too.
You live in a village, but it seems that the definition of village has been lost to all of your neighbors and so called governors. They simply do not have the right to be doing what they are doing, yet they keep pushing more and more. You have to say enough is enough and stop them.
Can they take away your animals? Not likely. Can they pass zoning ordinances banning chickens? That is the question I don't think that the state of NY gives village boards this power. A village is governed by the inhabitants. Your first duty is to build a fence and block your neighbor from property. You need to realize that you are part of the village governance. The so called board only exists with the rights given by the state, the rest is self governance by the dwellers of the village. You know like in friendly town meetings. Most of that history is probably lost, which is sad because you said this land has been in youe family for generations, and these people have you literally crying and begging to leave your own property. The only consoling factor in this should be they are so desperate because they probably realize they do not have the power to stop you from enjoying your chickens.