I know this is a little late, but I just came across the topic and read through the whole thing.
I'm an attorney in RI. Any attempt by someone other than the named party on the check, or his or her power of attorney, or estate, would be committing bank fraud. If you want the exact statute (here in RI) that this would violate, let me know and i'll post it. Here, it is a felony. Fraud, obtaining money under false pretenses, and conversion. It would be no different than someone other than me trying to cash my paycheck without my permission.
Tampering with the mail IS a Federal offense. However, this tampering also requires an element of "intent". Opening mail that has been erroneously delivered to your home on a continual basis does not constitute tampering. There is no intent to defraud, or commit a crime. That being said, opening the mail for investigative purposes, and then failing to notify the parties involved may look bad for you. Enough to arise to a criminal complaint? Probably not.
Here, the practical answer is as follows. You have no ethical duty to notify either the company sending the checks, nor the estate of the deceased. The estate should have done a more thourough investigation to discover assets, and the company made the efforts to deliver the dividends. The checks currently being held are likely no longer of any value. Most have either a 90 day, 6 month, or 1 year shelf life. The only criminal liabilty would be if one attempted to cash those checks. Asking the administrator or executor of the estate for a reward of some kind is neither unethical nor practical. If the party died 4 years ago, likely the estate has been closed, assets sold off (eg... the house), and any heirs have already received thier inheritances.
I would suggest contacting the executor of the estate and notifying them that you received some dividend checks for the deceased party. It is then up to them whether or not to pursue any debt the company may owe to the estate. The only benefit here , to you, is Karma... what goes around comes around. no good deed goes unpunished. Dont worry about any "criminal liability". There is none there.
If you have any questions, feel free to contact me
Steve