woodburning stove questions

Our new(to us) home has a red woodburning stove that sits on a raised platform which is covered in blue ceramic tile commonly found in a bathroom. The tile also covers the wall behind the stove. In order to deep six the ugly tile, must I remove the blue or can it be covered? Also, what's the best material to protect the floor and wall? Can I use natural stone?
Second issue is that our three cats tend to jump onto the top of the hot stove, and I haven't been able to find a protective cover to keep the stovetop cool. They're slow learners (with burned paws), but we can't get rid of them.
Any suggestions for either problem?
Thank you!

 

Remove the tile and replace it with whatever substance you like - rock, tile, brick.