Is there a free shuttle or bus from here to Hyatt Regency DFW airport after drop off of rental?

Yes, rental bus to airport terminal, then catch the bus to the Hyatt

Yes there is Robbie. After you drop off your rental car on the upper level in the rear of the rental car center (which is located at Passport Dr & Rental Car Drive) then stay on the upper level and go inside and you will see signs for the shuttles back to the terminals. The Hyatt Regency is located in Terminal C, so take the shuttle to Terminal C and you can either wait for the Hyatt shuttle or you can take the crosswalk over to the hotel (located at the 2nd shuttle stop gates 15 thru 25). Sincerely, Your Friendly Neighborhood Rental Car Shuttle Driver 😉👍