double fence and easement issues
My locale requires a minimum 6" set back from property lines for fences; check with your local building regulations and verify that a foot meets or exceeds the easement.

What makes sense to me is for neighbors to split the cost of a fence and be cool about it. If you want to tie into their fence and it is decent shape, get an agreement written up and pay them a fair price. If you can't work something out with your neighbor, then it does make sense to set it back a bit and create a gate so you can clean it out.

Cost is off topic here and it is too broad of a question. Check with two or three local builders and pick the one you like best.