No. This is an act of god. You should notify your car insurance.
Your landlord's insurance should take care of it. Legally your landlord is liable as they own the tree and supposedly should have had it checked and trimmed to prevent that.
It depends on what made the tree fall. If the tree is dead, then the owner of the tree is liable. If a tornado caused it to blow over it is considered a natural act and no one is liable.
Depends. Is the object a tree branch? Then possibly the owner of the apartment complex. If it is something that doesn't belong in a tree then it would depend on how that object got there.
circumstances are the key - health of tree? weather conditions? prior notice of problem with trees? if healthy tree in a windstorm or other natural weather event - nobody is at fault considered "act of god" you should contact comprehensive auto insurer
The driver
Probably not. If the tree falls on your house due to a storm then the neighbor is not liable for the damage and it will have to be filed on your insurance. The only case where a neighbor may be liable is if the tree was dead and the person knew it was dead and was a risk to your property. Why would you want to sue your neighbor anyway?
In general, Nobody is liable for an act of nature. Your auto insurance comprehensive coverage would cover damages to your car. Your homeowners insurance will cover damages to your home. The neighbor would be liable only if your can prove the neighbor knew the tree was rotten and posed a risk. If a car falls on your property, that would mean a tornado picked it up and it fell on your home. You insurance would cover that damage.
None, Trees fall all the time, It's a natural occurrence. Nobody is liable for an act of nature.
You will have to file a claim on your own comprehensive auto Insurance. A property owners insurance will not cover damages to your vehicle unless you could prove willful negligence. This is because a property owner is not liable for an act of nature.
Nobody is "Liable" for an act of nature. It does not matter if it was your tree, your neighbors tree or a tree that blew in from 8 blocks away. Your own homeowners insurance policy will cover your property. If your neighbor sustained damage, Your neighbor's Insurance policy would cover damages to his property. Just notify your home insurer of your loss.
Hopefully you have full coverage auto insurance with comprehensive. The tree owner is not liable for damage to cars from falling trees unless you could somehow prove that he was knowingly or negligently responsible for the tree falling on your car.