Sovereign Immunity is a doctrine whereby a state or sovereign cannot commit a legal wrong and is immune from civil suit or criminal prosecution. However, this doctrine applies only when the state or its agent is performing governmental functions.
You have to inform the bank of the arrangement you made with the other person. It would also be an advantage to hire a lawyer to help you with the mess.
Expect that your name would be dragged on the case, should a case be filed, because you owned the car. I suggest that you consult a car accident lawyer to help you with your case.
Yes, you should put signs. You should also talk to your neighbor to be responsible in informing their visitors about it. You can also leash your do because dogs no matter how trained they are can be vicious at times.