Will money Dad left behind automatically go to my mother?
Assets discovered after a death go to whoever is the legal owner of them. It's not clear who would be the legal owner of any bank accounts in your father's name that you discover. You state that you have not filed for probate. Did you father leave a will? If he did, whoever he named in his will to receive his property would be the legal owner. Often in a will, the will-writer names a "residuary beneficiary" to receive all will property not left to another beneficiary.
If your father left a will, you might or might not have to file for probate, assuming you discover any new banks accounts. Many states allow will property worth less (in total) than a set dollar amount to be transferred by affidavit or very simplified court proceedings. This set dollar amount varies by state, so you would need to check the law of the state where your father lived.
If your father did not leave a will, the legal owner of any new bank you discover would be determined by the law of his state. This is called the "intestacy law."
You state that your mother is mentally disabled. Does that mean that she is no longer legally in charge of her own finances? If she is not, then any of her husband's property she becomes legally entitled to would be managed by whoever handles her finances now.
