Want to make custom cage stack but need advice

[Argentra]

Sounds a lot like the 4 cage system I built long ago... so here are my suggestions based on experience:

What Kind of wood should I use to build this?
I use 1/2" exterior top grade plywood. @$12-13/4x8 sheet at Lowes

Should I use Wood Glue or will that be toxic?
To hold it all together? It may for a little while, but the best way to fasten is to use drywall screws. Wood glue can be used for external features like legs, but I wouldn't use it solely to put everything together.

As for heat I will use a 13.8" x 7.9" IntelliTemp Heat Mat on the inside of each cage, will this suffice?
Heat mats are plenty when used with a t-stat (that answer will follow). But since you can't just have the mat exposed, I use vinyl flooring in each cage over the mat placed on the wood floor with a few vent holes drilled in underneath. FYI I use ZooMed UTH pads.

What fairly low-cost thermostat should I use?
The best 'low end' t-stat is the Johnson or Ranco. They typically cost around $80 fully wired. Anything cheaper than that is a dial type and not to be trusted for full use. (In other words, they're only good for backups or QT)

What kind of lighting do you suggest? will warm temp white LED strips suffice?
I use closet lights for my cages, and my newest design will have them resting on a plexiglass window on the tops of the cages instead of mounted to the inside.

What should I use to keep the inside from molding?
For high humidity snakes, like BPs, I line the inside of my cages with vinyl. The floor is usually full sheet pieces, the 'ceiling' and sides are tiles, and the back (as I like using scrapbooking paper as backgrounds) is paper covered in clear 'tablecloth' vinyl sheet pieces. For other reptiles I simply paint all wood, both sides, with a latex based paint that is wonderfully water resistant.

Using these materials doesn't cost too much either... trust me, I've always had to build on a budget.