I have my humid hide on the cool side of the cage. I use both vermiculite and moss, depending on the leos really. On two females that makes a heck of a mess with moss so I switched its hide over to vermiculite and have not had a mess at all since from the two ladies
I'm pretty sure the humid hide should be on the warm side. For what is inside it, I just took some folded paper towels and placed some ReptiCarpet on top of it. It doesn't stay moist for a very long period of time, however. I'm currently trying to figure out how to fix this...
Warm side-cool side-in between. It is up to what your gecko prefers. If you are looking for something to serve as both a moist hide and lay box I would recommend coconut fiber for the material. It holds moisture well, and the girls can dig easily to lay their eggs.