Legally you can remove a neighbor's attachments if they are on your property. You might want to consider mediation because disputes like this can escalate into problems.
As long as the fence is on your property, no.
You first have to prove this fence is in fact, on your land and not your neighbors. You do this by obtaining the survey that is filed along with your deed. And then, if there is further questions, you might have to hire a survey company to re survey your property, physically mark the property lines and then give notice to your neighbor that his fence is on your property, justified by a copy of the survey and give him a time frame in which to remove this fence. Then, if it is not done, remove it and sue for the amount it took to remove it. You should read the contract you signed when you purchased the property. Many contracts for the purchase of land will state if an existing fences is the property line regardless of what a survey shows.
The fence cannot be outside your property line.
Yes, if they own the fence.
If the plants are on your property they may be planted along your neighbors fence. However, climbing plants that will grow over the fence cannot be planted along your neighbors fence. Caveat: Make sure your neighbor has not installed the fence "inside" their property line so it lies a short distance FROM the property line. That is the practice in some areas and is required by the city ordinances in some areas. In that case, the land along the fence would belong to your neighbor. You need to determine where the property line is located and not assume the fence is on the property line.
My neighbor would like to remove the fence that we share in our backyard. Is he allowed to remove the neighboring fence without my approval?
Build a fence. It is not uncommon for land owners to fence off their property to keep people or animals out.
when building a wood fence in the town of Brookhaven what side of the fence is supposed to face my neighbors property
If the fence is their property, and not yours; you would need their consent to make any modification to it.
A common fence is a shared boundary structure between two neighboring properties. It is typically built and maintained by both property owners to provide mutual benefit and to mark the separation of their land.
If the neighbor's fence is on your property in New Mexico, you can ask them to remove it. If they refuse, you can take them to civil court.
No