Animal Control: Normally I would say that is too far because if someone reported your human babies for crying, you would be pissed. And dogs do bark. But it sounds like you exhausted every other measure before getting to Nuisance Barking level. To be honest though, I would have dealt with the barking. Then again, I prefer dogs over babies and can definitely deal with barking over babies crying. But to each her own.
As far as you joining the board, hooray. You joined the flashlight cops. They DEFINITELY could have allowed you to unpack before this fence issue came into play. My guess is the prior owners kept putting it off, said you would do it and the HOA was like "I am DONE asking." For them to come at you on Week 1 is a bit much though. Nonconfrontational = Unproductive HOA president
Anyone in leadership has to be ready for a challenge. But it is wild that fences were his final straw but loud, barking dogs were OK. At some point, my dog is going to think her name is “Shut up" if she keeps barking at 7 a.m. though. She is waaaay better than in Month 1 though. There should be a settling period. It sounds like her dogs may have separation anxiety. I play Lo-Fi music when I leave. My dog barks her head off when I leave and then it is so quiet after the door closes and when I come back. I almost think something is wrong, but she is just hanging out in her crate watching TV. The barking just completely stops.