Geckos normally need to be misted in captivity because they are dumbdumbs who can't understand the water bowl and normally drink the little droplets that collect on the decorations in their tank. So long as it has decent enough humidity and is misted it should be fine.
If you are keeping it in something I would encourage you to let it be free back in the wild, but that's because I believe if it's a wild animal it should stay one. If you don't want to let it go I understand, they need lighting so if it is just roaming around the house i would put get it a viviarium (they need a tall cage rather than long), good substrate to keep the moisture in, as well as some things to climb on. Idk where you live but it's going to start getting chilly soon so I wouldn't let him just roam around for too long just in case.
On the bug part, feeding dead/wild isn't a good idea. You're technically supposed to be feeding gut loaded insects to captive lizards, "Gut loading is the process by which an animal's prey is raised and fed nutritious foods with the intention of passing those nutrients to the animal for which the prey is intended", since it is wild a wild insect wouldn't hurt but I would skip the dead mosquitos asap, either wayfeeding wild can give it some nasty parasites. Online it says house geckos likes crickets, mealworms, waxworms, silkworms and roaches. You can get the first three at pretty much any pet store but silkworms and roaches are harder to find. For geckos I'd suggest the wax worms since mealworms shells tend to be a bit hard on digestion. You also have to keep his bones strong by sprinkling his feeders with calcium powder. The rule of thumb is to look at the space between the eyes which can be used to determine the largest size it can handle as far as feeders are concerned.
Hope that helps lol, I didn't really know how you were keeping it so I tried to include everything I could think of.
Edit: should you return it to the wild do it where ever you caught him, if not you'd only need to do the calcium things three times a week at most. If you want more advice pm me and I can give you any info you need. I only have a crested and a beardie though, so any house gecko info will be found online.