What is the rule or common practice about when an account hits a zero balance and considering it closed? If the customer comes in days later and says they didn't mean for it to close do you allow them to deposit and reopen without basically treating them as a new customer for CIP purposes?

If your depositor's contract stipulates that the account will be closed at the occurrence of this event, then I believe the account is legally closed. If they have no other active accounts with the financial institution, it would be a new account.

Ok, thank you.

A little off-topic, but if the Bank closes an account due to suspicious activity or fraud, must the Bank return the funds in the account to the accountholder?

Do the funds belong to the account holder?