Unfortunately home Depot will not cut tile for you or Lowes. You'll have to find a way to cut these tiles. Ceramic tiles are easy, score and snap like glass. Natural slate......you will have to cut with a tile saw. Good luck with it.
Well, I think it depends where. My home depot does not cut, but Lowes does. So you might have to ask around and see who does what. If you can get the tiles cut really precisely then you usually don't have any gaps to worry about, and if you do you can pack something like a strip of cork into it or something.
The type of tile is up to you. I prefer natural tile because it's easier for your geckos to grab on to without slipping and sliding. But as long as it's not too smooth it's really up to you.