You don't have to give that persons a key, but he or she can access the box after the death of the box holder without delay because the person is on the signature card. Of course, the key would be needed at that time. In the county of residence of the deceased. This is the legal residence, not the county where the person might be at the time of death. In that event, the process is called administration.
Some new procedures make that a much easier process also.
A separate page exists on this web site on the subject of administration. Not at all - not if there is a properly drawn will. One of the standard provisions of a properly drawn will is one relieving the representative of the duty to file an inventory. No listing of assets is filed in court. Nobody will be able to know what the properties are.