I consider using them as gym flooring and assume you don't recommend installing them directly over medium-pile carpet. Do you recommend putting a layer of plywood over the carpet first before installing them? What thickness plywood should we use? I'm not sure whether it's better to install a 3/4 inch plywood and then install a thinner rubber mat or use a thinner plywood and use a 3/4 inch mat?

You can install our 3/4" Extreme Mats over low pile carpet with ease. Over a high pile, thick padded carpet, we recommend placing 1/4" plywood boards over the carpet to prevent separation of the mats. If you are installing over a medium pile carpet with a thick pad, you will install the mats as you would over a high pile carpet over a thick pad. If the pad is not very thick, then you wouldn't have to use a plywood board.