There are over 100 tennis courts at HH. The resort does not have it's own, but does have an arrangement with the nearby Palmetto Dunes Racquet Club (it's on the way to the Beach House). They have hard and clay courts available.
The resort health club is a small room with a few exercise machines (treadmill/stepper/nautilus) available (as well as a balance scale!) It was refurbished this summer.
Enjoy!