Like Jeremy said, ditch the acronyms.
Should I cover the inside of the cage floor with another material? Should the bottom be of a totally different board?
I would definately cover the inside of the floor with some other material. Scrap Sintra or other expanded PVC found at sign shops works very well as does vinyl flooring. Fiberglass reinforced panel (FRP - see, a defined acronym) works well but is a bit harder to find as scrap and a bit harder to cut.
If you must use an under tank heater (UTH) I would not place it inside of the cage. Instead I would cut a rectangle out of the floor of the cage that is 2" longer and wider than the size of the UTH that you plan to use. Make sure that cutout stays away for the walls of the cage so it maintains structural integrity. The flooring material can be installed directly over the entire floor and the UTH can be taped underneath the flooring and inside of the opening. This provides an air spaced and makes the cage safer.
Can I stain it?
You can stain the cage. Understand, however, that staining birch plywood, especially the lower quality stuff from Home Depot (if that's where you got it) does not always yield the best results. Birch in general does not accept stains well and birch plywood is even worse. Compounding this problem is that for a reptile cage you'll probably want to use a waterborne stain which can be very blotchy.
Personally I like the look of birch ply with just a clear coat. If you do stain your cage you might look into an aniline or other type of dye that can be mixed in with a waterborne polyurethane.
What do I seal the wood with?
What do I seal the cage with?
Isn't this the same question since the cage will be made of wood?
Regardless, if you do use a plastic floor then I believe that a waterborne poly is sufficient for the walls of the cage. I would avoid the oil-based polyurethanes. They can offgas solvents for years and are not even supposed to be used in drawers that hold your socks much less a reptile cage.
Of the readily available waterborne polurethanes, I like the Varathane Diamond Poly for a plywood cage, not the Diamond floor or spar finish. If you live near a woodworking speciality store there are some even better WB polys.
What is suggested for the door? I'd like to do a framed door that is a hinged door. I don't feel like messing with sliders.
Sliders are much easier to build and generally considered to be more attractive due to the more open look. I prefer how they function compared to a hinged door but if you prefer hinged then that is what you should build. I think the easiest way to do this is to build a frameless hinged door from acrylic, sort of like what you see with Boaphile and other cages. But this is more expensive.
Lastly, you can cut your ply better than Home Depot can. The blade on their panel saws will chip birch plywood pretty bad. Just invest in a good blade and edge-guide for your saw. A carpenter's square will be useful as well.
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Current snakes:
0.0.1 Gonyosoma oxycephala - Java locale (green)
2.2 Gonyosoma janseni - Seleyar locale (all black)