Didn't know if y'all know, but we breed & sell leopard geckos (I have a bunch of eggs in the incubator as of right now, as well!) Many different "morphs", but there are literally hundreds of morph combinations out there.
We keep all our geckos on paper towels or newspapers. Change once a week and they will not stink...ever. Your lil guy is NOT too skinny...he's just fine for his age, which I would put at roughly 6 mos. He will continue to fill out as he gets older & have a very fat tail in another 6 mos.
Definitely keep a moist mossy hide in there at all times. Most of ours spend 90% of their time in the moist hide and only come out at feeding time. We also use ONLY orchid moss, as the cheaper stuff molds quickly & is a pain to keep clean. You can utilize a 16 oz deli cup with an entry hole cut into the lid as a moist hide to keep the moss in one place. We also give them regular hides, with no moss, but they tend not to use those.
We also do not use supplemental heat, as they don't need it unless you are keeping your house VERY cold. (i.e. 70 degrees or less) And they absolutely do NOT need a "basking" lamp. They are nocturnal & live in the deep rock crevices in Pakistan & northern India, so they do not naturally get exposed to a lot of light/heat.
They DO need a grit substance which they need to properly digest food (like birds). We make ours with clean sand, ground egg shell & calcium supplements (which they need to prevent metabolic bone disease.) We sell this premade, but you can mix up your own.
In order to sex your gecko, you may need the assistance of a bright light & a magnifying glass. He looks like he *should* be big enough to sex without the magnifying glass, but it can definitely help. Turn him over and look near the "vent" area. You should see a 'V' shape of scales there. If there are actual PORES in those scales, your gecko is a male. If there are just slight indentations or no pores at all, it is a female. If you are having difficultly, see if you can get someone to take a pic of this area (on macro setting) while you hold the gecko on his back...and I can look for you.
As for food, he'll eat crickets, mealworms, roaches & smaller superworms right now. I just put the mealworms in the dish of calcium grit, so that they get coated before being eaten, but you can also dust roaches, crickets & worms in a baggie with calcium, too. They need a LOT of calcium, so dust all the time! (Use NO D calcium...they don't need Vitamin D, and too much can kill them.)
Let me know if you have any questions at all, and congrats! These guys are one of my favorite geckos!