Here is a video of me riding in the back of a horse trailer at 65 Mph down the road. You will see and hear what a horse hears. I do not put shaving in my trailer since I don't want it blowing around and getting in my horses eyes, ears and nose. I normally don't feed or have hay in the trailer for the same reasons.
Floor rubber mats are a must as far as I am concerned. They distribute the weight of the horse, they stop road debris from blowing up under the wood planks, they stop water and oils from being splashed up into the horse, they keep noise down and cushion the ride for the horse. I have seen many people just load horses on the wood and in an emergency that is OK, but for routine hauling, it is not fair to the horse and is shows poor horsemanship.
By taking a ride in your trailer, you can spot rattles, you can hear, see and feel what you horse feels so maybe you will be a little more careful and a better driver when you are carrying livestock or horses.