Excellent questions
I used sheet metal screws with a flat head. The outer shell of the fridge is sheet metal and the inner skin of the door is plastic, these screws work well for both. I predrilled both the plexi(slightly larger than the screws) and fridge(slightly smaller than the screws).
The eggs actually site on top of the egg grate, not touching the Perlite. The Perlite is really only there to hold up the egg grate and keep the water from sloshing around when moving the egg boxes.
Excellent point that I forgot to mention! I DID use small water bottles at the bottom of the incubator to serve as a heat sink to help stabilize temps when opening and closing the incubator.