I have a front loading washer & dryer and have done this (vacuum them thoroughly first). One little tip...sometimes the dryer heat causes the rubber backing to start crumbling off, so don't use high water or dryer temps. Since making that particular discovery I hang them to dry using skirt hangers (they have clips for holding them without having to bend or fold them).
Sure you can stick anything that will fit in, TVs, kid brother, dog, computers, microwave, you name it. I wouldn't put the mats in my washer & dryer, many laundry mats have heavy duty commercial machines for big nasty stuff.