If it's on your property or his actions caused damage to any of your property.
If the fruit falls onto your side, I believe it would be yours. However, though some localities say a homeowner can cut overhanging branches IF the cutting does not deface the tree, it is MUCH more neighborly to work out the problem with your neighbor. No point in hurting a tree, especially one old enough to produce fruit.
woodcutter
The tree that has pods hanging down from its branches is called a mesquite tree.
If your neighbor refuses to cut down the tree, use insurance or savings to cover taking down your fence (and putting up a new one), get him/her to sign a paper that says he will hand his tree over to you in exchange for a portion of money from, then put up the fence (or wall, in this case), and do with what you will to the tree. If that doesn't work, move, or trespass property by cutting down the tree, and risk getting sued.
Legally, the branches of a tree that overhang onto your property still belong to the neighbor. You have the right to trim the branches that cross onto your property up to the property line, but you must ensure not to cause harm to the tree itself. It is best to try to discuss the situation with your neighbor and come to a mutual agreement.
If possible, you should request that your neighbor trim the branches. If you trim them and somehow the tree coincidentally dies, there would be further problems.
Vibration of WHAT? The noise of the saw? The impact of the tree when it hit the ground? Etc? Unless the tree actually struck your residence, proving damage to your drywall is going to be difficult if not impossible.
Yes, cutting down a tree will eventually kill the roots because they rely on the tree for nutrients and support.
Because any chemical changes occur.
Cutting a tree down does not necessarily kill its roots. The roots may still be alive and continue to grow even after the tree has been cut down.
I would imagine yes but will depend on your local laws. Phone your local council for advice. In most municipalities, the neighbor has the right to trim any branches off your tree that hang over his property--at his expense. Insurance companies expect that homeowners will protect their homes from any tree branches whether they belong to the homeowner or the neighbor's tree
Felling