1, depends on the place, but a standard disclaimer should get you out of needing insurance.
2, yeah weve had corners of floor mats rip off and had horses rip, they dont know about it yet i dont think, but as the hirer im responsible for any cost due to damages .
also cuff bar bolts can easily pop into mats, big crash mats are fine, but floor mats hate them.
3, yeah, our last gym meet had loads of skid marks from rubber feet, but the cleaner had told us not to worry about the mess, an old moist towel (the one you sweated on most of the gym meet is perfect) thrown on the floor and then worked with a foot will get any skids up pretty quick, if they are funny about the floor thats an easy way to leave a "clean" floor.
as for scratches etc ive never seen anything like that happen, seen as your in a gym 90% of falls will be onto mats in my experience