Barring some provable negligence, Nobody is liable for an act of nature. Your home insurance should provide coverage for a natural hazard such as this, Likewise if your tree fell on your neighbors house, the same would apply. The neighbors insurance would take car of the neighbors house.
Depends on a lot of things. Generally, the tree is considered part of the land. Where it falls is who is initially responsible for the damages. From there, it gets legal depending on multiple factors.