My name is ***** ***** I am a NSW Solicitor. Thank you for your question, and will do my best to assist you with your question. Please understand this is not legal advise Please understand this is not legal advice but a guide to assist you.
Have you had an arborist look at the tree and advise if it is dangerous and also have you had a plumber confirm that the roots of the tree are the issues?
What you are describing is nuisance and he is liable for the mess that comes from his tree.
You have a right to cut back the branches that overhang on your land provided you do not kill the tree.
But being such an old tree you should first seek council permission because they may be protected.
Here is some information that covers trees and all other issues between neighbours.
You have rights and he cannot simply refuse to do nothing and you have to accept his refusal.
The above 2 links will assist you to understand your rights.