Yes, AA Flagship First Dining is only for AA marketed and operated flights. However, you would be able to use the AA Flagship Lounge in both ORD and DFW. Having been to both in the last few months I can say they are very good.

When you land in ORD upon arrival from LHR you will need to clear immigration and customs first, then change terminals and clear TSA before accessing the lounge ahead of your onward flight to West Palm Beach.

As you're Gold / OWE you would have access via status irrespective of class of travel (although as you are travelling F on the long haul sectors you would also have access based on class of service).