Your neighbor is responsible for any work he asked a contractor to do. If he wants to recover the cost from you and you don't agree to pay, then he will have to ask the judge. The judge could go either way on this, you never know with judges!
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All the parties who share the common fencing are equally responsible for it's repair or replacement. The owner of the property on which the tree was located is only responsible for their "fair share" not for the entire amount.
In all likliehood - your insurance company would pay to repair your dwelling and then they would take action (called "subrogation") against your neighbors insurance company to collect from their insuror.
If he accidentally fell in and tore it, it's not his fault but the dog's owner can be held responsible to repair the damaged liner.