First of all, you have great timing and you caught some great action shots! You set up in the Gym which is a great idea. I set up in the ballet studio...same thing. I have gotten away from cloth or muslin backdrops because of the wrinkles. Except for black. Wrinkles are there in black but you cannot see them. I use 12 foot wide paper on a roll. I have a black and a white roll. This gives a nice background.
I also use softboxes and my subjects are usually 5 to 8 feet from the backdrop. This kills the shadows on the background. I take front cloth off the box to get shadows.
I did not like the floor so I got a real section of vinyl ballet flooring called "marley" it comes in 5 foot wide rolls so I bought 32 feet, cut in half so I have a 10 x 16 floor.
If I were to shoot these gymnists, I would use a 12 foot white background and tape it to the floor (use clone tool i PS to get rid of tape)so that the real wood floor was "the real floor. The equipment would look more natural this way. I would also try a black background with the wood floor. You need the 12 foot width for the equipment.
As for shots, you have the opportunity to do things no photographer could do in a competition. I would sit the camera on the beam and shoot straight down the beam at the gymnist. Same with parallel bars. If you can control things, place the camera to capture angles and expressions you could never get in competitions.
I use at least 3 lights. Also Photoshop is very useful.