I have heated ceramic tile floor I still use a bath mat for stepping out of the shower but no rugs elsewhere. When I wash the floors I notice that they dry much faster than non heated flooring. So if you don't use a mat it would dry faster than otherwise but I think you might still want to use a mat to catch most of the drips.

I like a shower mat to dry off the bottom of my feet, not to keep them warm. It provides traction, too.

Rugs would not be needed to catch the drips. The wires are sealed. Unless you have a problem with cracks in your grout or tiles, I wouldn't worry about drips. Most installers don't install the radiant fabric all the way up to and around the toilet and up to a door threshhold for example. You can pretty much feel where it stops. So if your toilet leaks, for example, shut off your radiant heat just to be on the safe side until it is repaired. We've had ours 13 years and never had a problem. We use a rug, like badgergal and weedyacres, just for stepping out of the shower...nowhere else.

If it's the combination of water and electricity that you're worried about, there's really no need for worry.

As others have mentioned, the wires are fully insulated. To go a step further, there are manufacturers whose products are approved for installation in showers, beneath the tile but above the waterproofing membrane.

Now all installations can be messed up. But if your RFH was properly installed and properly wired, and your floor structurally sound, you should have no worries.

A quick word regarding RFH and toilets. A common reason the RFH should be held a bit away from the toilet is so the heat doesn't warm the toilet's wax ring enough so the wax can deform and cause a leak. I'll usually change to a waxless ring.