I would go with slate tile, looks great and natural, and retains heat very well. Reptile carpet tends to harbor bacteria and can snag geckos' claws and teeth.
I don't believe so. What substrate are you keeping him on now? As for the slate cleaning question, I cannot answer it, due to the fact that I use paper towel. However, many people here have lots of experience with slate tiles. I'm sure someone will chime in with an answer.