Is a bank required to cash govt. checks (US or State) and cash US savings bonds for non-customers?

Checks, no, unless you are subject to a state law requirement for cashing Social Security and certain other checks for recipients of government benefit payments. If you are an authorized redeemer of savings bonds, you have to follow the instructions in the bond guide. Without looking it up myself, I believe you have to redeem bonds for non-customers up to $1,000 with acceptable identification.

How do I find out if we are an authorized redeemer of savings bonds?

John...in redeeming up to $1,000 of savings bonds for non-customers with documentary identification, are we still required to perform an OFAC search on that person?

Replaying broken record: There's no requirement that you check anyone ever. There is simply a prohibition against doing a transaction with a listed party. Your bank should have a risk-based policy to address your question.

Easiest way to figure this is if you sell them then you must cash up to $1000 for noncustomers.