As long as he is in a room with a window to the outside he is fine as far as light cycles go. If you want you could put a lamp on a timer to click on and off as well elswhere in the room.

I wouldn't cover the entire top of the enclosure. Should be ok with just removing the light you have. Your cage is only drying out because of that lamp, normally you only need to spray once nightly to achieve proper humidity so I can tell that the light is the main issue if you need to spray every few hours.