Live in a twin what? Do you mean a duplex?
No. In fact no part of your neighbors roof or anything attached to that roof can even protrude over your fence. Most places have laws specifically stating how far away from your property line they have to keep their buildings.
(If you can prove that it is more likely than not that the neighbor dumped the fence panels): Hire a licensed contractor to remove the fence panels. Then, take the offending neighbor to small claims court for the amount of money you paid the contractor to remove the panels.
It is not possible to attach to a neighbor's fence unless permission is given by the neighbor. It is a good idea to get any agreement in writing for attaching to the fence.
If the fence is yours - ie your boundary and on your land then the neighbor has no right to attach anything to it (without your permission).
If the grapevine is growing onto your property, you can trim that growth back, but you cannot destroy anything that is on your neighbor's side of the fence. You must also offer your neighbour the prunings back as they belong to him/her.
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?
If you can prove it your neighbor should be responsible for it, through insurance or otherwise.
Perhaps you have mixed up your words in your question. It makes sense if read as, "If you painted your side of your neighbor's wood fence can the neighbor make you replace it with a new wood fence?" If that is the case, you had no right to paint your neighbor's fence without their permission and the answer to your question is they can sue you in court for damaging their property.If they sue you in court and prevail, they have the right to be restored to the position they were in before you damaged their property. If the fence is new the court may award enough damages to cover a new fence. If the fence is older, the court may pro-rate the value. While your neighbor cannot make you replace the fence, the court can and you may have the added burden of legal expenses. You should try to negotiate a friendly agreement to resolve the issue either by repainting the fence to please your neighbor or perhaps offering to share the cost of a new fence.As a lifetime lesson, it is more prudent to let the neighbor know your intentions, even if it is your side of the fence, or if you think you are doing them a favor.
Privacy fences don't have to be of the same design throughout the yard or garden. Different types of fences can delineate different areas, or "rooms" of a garden. A simple picket fence can cordon off a vegetable garden, while a painted metal fence connected to stone gate posts can border a private space in the garden with a stone bench and a fountain or pool. Even a chain link fence used to separate the lawn from the neighbor's property can be esthetically pleasing if it's painted a color that blends with the rest of the landscape. A rough, natural fence can blend well with natural surroundings and may be used as the border between the cultivated areas of the property and the woods beyond.
The perimeter. It is the total length of the boundary (where the fence is or will be) of the garden.
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.
That type of issue is governed by local building ordinances. You need to call your town clerk and inquire about fence regulations.