Yes, in many states a garage is responsible for providing reasonable security for vehicles stored on the premises, not to mention security for people who are going to and from their parked cars (lighting, surveillance, etc).
Furthermore, even if your "claim check" has something about "not responsible for damage or theft to vehicles", it may not be sufficient defense for the garage. In some states it is illegal to even try to enforce a waiver of liability like that and doing so is an "unfair and deceptive business practice" amounting to fraud.
The rules on open-air parking are somewhat different.
From being parked only - no.
Garage
Yes.
Yes. The car created a traffic hazard. The person who parked the car is responsible for any accident that occurred as a result.
The one backing out, because you should realize someone is opening a door.
The owner of the illegally parked vehicle may be ticketed, but the "at fault" driver is responsible for the damages to the parked vehicle.
your car cannot be reposessed if it is in your own garage
In brief, if they can get to it, yes they can.
You are responsible. You should not have been parked illegally.
The position that can safely be used if the car is parked in a closed garage.
If they were not parked illegally then they are not responsible for your accident.
I think it would have to be the person who hit your car because they caused your car to crash into the other car.