Yes, in fact filing suit is often a requirement, to prevent the running of the statute of limitations on the trespass, after which you would have no way to evict them.
Simply being the owner of the tree does not make you liable. Generally, No one is liable for acts of nature. So if the tree falls due to a natural occurrence you are not liable. If however you were...
Yes if they got hurt due to your negligence you are liable if they get hurt and it's their fault then you shouldn't (depending on the incident) be held responsible. ...Known from experience
Depends on who they get in the fight with. They can always sue. Whether they will be successful or not will depend on the situation. If two people with no connection with the homeowner get in a...