Not without your permission.
His gate-yes. Just as you can trim tree branches extending over your property without permission even if the tree is planted on someone else's property.
Yes, it is illegal due to the dog is not certified to you.
Technically in most states, you cannot enter his property without his permission. It is best to seek mediation or get the neighbor's permission before going on his property.
If you borrow your neighbor's car without their permission you are a thief. So if you borrow your neighbor's Internet connection without their permission you are a thief
In most countries, you should not (without permission) trespass onto your neighbor's property to cut the tree, however it is quite in order to trim any part of a tree that overhangs the boundary of your property (up to the boundary only). It is also important that you return all the wood that is cut off to your neighbors property (it is his tree).
How well do you and your neighbor get along?Legally, I think you are tresspassing if you don't have permission to do so. Just ask if it's okay.
They have no right to change the location of the access easement without written permission from the property owner.