Welcome to permies Jane!
Posting a diagram of the
land would be helpful.
1. My first thought about the treed area is to make a woven fence through that area using the brushy wood that's already there. Whatever those trees are, they seem like a coppicing/thicket-forming type of tree that would respond to a variation on the "living fence" concept.
Not sure if I would get a permit for it...but I would try.
Do you mean that you need a permit for any sort of fence you erect? If done subtly, what I'm picturing in the woody area might not actually "look" like a fence, thereby possibly not require any sort of permit.
2.
30% back up to neighbors goat area with hog wire fence. I need to block goats out or my dog would go crazy.
My thoughts about this: if you're just trying to block the "view" of the goats? If so, I'd use plants to accomplish that - preferably useful plants. My example - years ago I moved into a new subdivision when everyone wanted solid 5 ft fences with a ft of lattice at the top. They all quoted "privacy" at their reason, but I knew that they'd been convinced of that without thinking it through as many of them were 6' men who could see through that lattice, whereas as a 5'4" female, I could not see back - or even easily see them watching!!! I refused to pay for their folly, which was my legal right. Instead I put in a small patio near my house, put up trellises around it, and grew scarlet runner beans, which quickly filled in and gave me privacy where I needed it (and dinner too!) One of those neighbors who had chosen to build an elevated deck, allowing him to sit down and still see over his "privacy fence" complemented me on how smart I'd been and how he wished in retrospect, he done something similar!
3.
Fence needed to keep dog in
Have you considered that buried wire type dog control system? Is this dog allowed out unsupervised? How high can said dog jump (or is he a digger)? This is where a map of the land and building would be useful. Here on permise we often talk of "paddock-shift" systems where animals are give different areas to use in succession. I've used dog exercise pen fencing as temporary corrals for ducks all the time. That way I can shift them to fresh grass and let their old paddock have a break.