Is there a requirement that the name and city of the paying bank be printed on a checis that we have printed for customers?

City - no. I think printing the bank name is customary. But for specifics: O.C.G.A. 11-3-104

Many years ago I worked with a bank that changed from a state to a national charter. At that time check printing was changed to show just the name of the bank, website and 800 #. And then I had a national chain refuse my check because the bank address wasn't printed on the checks. I don't recall why I wasn't using my debit card, so I had to pay with a credit card, and I was not a happy camper with that merchant. But they are completely out of business now, so it's hard to hold a grudge.