Many moons ago, when I had my 2 horse straight load, I pulled my mats up if the trailer was going to sit for more than a few weeks or immediately if there was moisture/pee – but I was a kid, with her first trailer and the floors were oak. Yes, the mats were flipping heavy, but a friend taught me the trick of drilling a hole in the end of the mat and using a hay hook to drag them around. Made a painful job slightly less painful.
If you’re in a dry climate, don’t have grass under where the trailer is parked, and use the trailer somewhat regularly, I probably wouldn’t bother every time.
I only used a dusting of shavings when traveling, less than a 1/4 of a bag for both horses, just enough to offer traction if someone peed and the mats got slippery.
Always cleaned and swept the trailer after hauling, I never left shavings or hay or any organic matter in the trailer. But I live in a wet, humid climate where things mold right quick.