For a home gym, 15.5' x 13.5' that will have a weight lifting rig, assume some reasonably heavy weight dropping, 300 pounds plus, do you suggest 1/2" or 3/4" in mats? My CF gym, which is a very large space has 1/2 inch stall mats.

Our 1/2" Premium Mega Mats are plenty for this type of set up. These mats are vulcanized, meaning they are more durable that typical stall mats. These mats are some of our most durable mats and were engineered for heavy Olympic lifting. 3/4" is not necessary for most home gyms. The purpose for a thicker mat is really to protect the subfloor more, say if you have a plywood subfloor, and reduce the amount of noise transmission between floors. If you are installing these mats over concrete, then you won't need anything more than 1/2," honestly.