There's no D1 service between ATL and MSP even on the A330 with flat bed seats. It will be regular domestic FC meals, wine, etc. Only a few routes are designated as domestic D1, IIRC only between LAX/SFO and JFK currently. Domestic FC doesn't entitle you to lounge access.

International FC might, IIRC it's true for the Canada transborder flights, but I don't think DL considers Nassau to be international for this purpose.

There's a useful dropdown menu on the DL website that indicates which lounges one can access under what circumstances.