What types of checks are banks legally required to cash for non-customers? In particular are we required to cash U.S Treasury checks? And we are not required to cash postal money orders, correct? Any guidance would be appreciated.

You are not required to cash any of the checks that you mention. The only checks I believe that you have to cash are on us checks, that are properly endorsed where there is sufficient funds.

And ..... You need not cash on-us checks until the end of the business day -- giving you some opportunity to verify the validity of the check

David is correct, there is no federal law that requires you to cash any type of a check for a non customer. There may be a quirkie state law; e.g. I know of one that requires banks that are state depositaries to cash state issued checks. In effect, it's an enhancement of the "on-us" concept.