1. You can't unknow what you know. In other words, regardless of you having a copy of the Death Certificate, the POA is no longer valid if the customer is deceased. Your policy and procedure may require a copy of the Death Cert to do anything further with the account, but I would not let a POA agent transact on the account if I know that the customer is deceased. Also, the statute states that actual knowledge of termination of the power (i.e. event of death) is grounds for refusing to honor the POA.
2. We do new signature cards when adding anyone to an account. Check with your policies and procedures, but I don't see why you wouldn't.
3. I would think this one is a matter of your bank's policy as well. If the POA agent is not a person you would establish a relationship with, then you can decline to do so even in a POA situation.
Opinions are my own.
"I was gratified to be able to answer promptly. I said I don't know." - Mark Twain