Some neighborhoods restrict the type of fence you can put up, say if you have an HOA or live in a Historic neighborhood, so check and see if you have restrictions. If none, look at the survey of the property and make sure that the fence posts, if you are doing the work, are on your property. I know of many people who have thrown up a fence without a survey, only to learn they or the CONtractor planted the posts on the neighbor's property. Oops....
Ask the owner of the property next door if you can trim the trees/branches. Legally you don't have to, but it is a good thing to do, so that in case a branch falls on your house and causes damage, you have a relationship so you can file an insurance claim. I would recommend you outsource the tree trimming to limit your liability, use a licensed, bonded and insured tree service. Only trim the items that are over your property line, unless you receive written permission from the owner. Make sure to get it in writing, because we all know that no one changes their mind.
Best of luck to you,