A few states used to have laws requiring that a box be sealed temporarily on the death of a lessee (individual or joint). I know the number is dwindling, but check you state law on that score. It has something to do with taxes.
On your second question, a safe box is not an asset nor is it an account that can be transferred on death. At least one state (Iowa) has a law allowing a bank to permit the naming of an individual who can access a box after the lessee's death, but that's strictly temporary, designed to allow for will retrieval, etc.
The contents of the box become the property of the estate, and the lease continues in the name of the estate. Many, if not most, banks will require that a new contract be signed by the estate's representative if the box will be retained for a period of time.
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John S. Burnett
BankersOnline.com
Fighting for Compliance since 1976
Bankers' Threads User #8