Hi Benjamin,
Thank you so much for visiting the Disney Parks Moms Panel with your question.
The Villas at Disney's Wilderness Lodge is a lovely resort. The theme there is calm, rustic, and relaxing.
There are five resorts that have
basketball courts at Walt Disney World. The link I provided lists both the resorts at which they are located, as well as the type of court at each. There are not basketball courts at The Villas at Disney's Wilderness Lodge. The closest basketball courts to The Villas at Disney's Wilderness Lodge are at the Campsites at Disney's Fort Wilderness Resort. I called Disney's Wilderness Lodge to determine if you would be able to use these, and I suggest that you ask at the time that you would like to play if this is possible. Asking a recreation cast member at Teton Boat & Bike Rental is probably a good idea.
As to the fitness center, at Disney's Wilderness Lodge this is the
Sturdy Branches Health Club. According to the site I linked, guests must be 14 years or older to use this facility. Guests who are 17 years old and younger must be accompanied by an adult to use the facility.
Have a great time on your vacation Benjamin, and please do not hesitate to ask if you have any further questions!