Most “cash discount” programs are non-compliant, because they list the cash price on the shelf and then mark it up at the point of sale.
When you add a fee to a listed price, you must comply with the card brand rules, and must not apply the fee to a debit card.
A compliant cash discount would show the “credit card price” alongside the “cash price” with equal prominence whenever prices are posted or quoted. See a compliant example.
Almost all “cash discount” programs fail to meet these requirements, risking fines from the card brands and state attorneys general.
No, you cannot.
Yes, they are eligible for the “service fee” program.