I like cage stacks where the floor of one cage also acts as the ceiling of the cage below. I'm getting ready to build some myself.
However, your cage seems like too much of a good thing. One thing off the bat that I don't like is cage this is 12" tall but 36" deep. How are you going to reach in the back to reach a snake or to clean the cage? Remember, you'll likely need some sort of lip or face frame which means the actual opening will be less than 12", unless you already accounted for that.
I would say make the shortest cage a taller and decrease the depth. I think 24" deep is fine and saves on material. 30" is a max.
Besides, how would you move a 48" x 36" x 72" into your house? The original builder of my house had a handicapped son and it is built with wheelchair access in mind. I could not get a cage like that from my garage into my house, that's for sure.
Don't groove the wood for glass. Instead use plastic glass track glued to a face frame of some sort. You will absolutely need a face frame to add structure to a cage like that. It will also act as a substrate dam and give a finished look to the front.
You could use either 1/4" or 1/8" glass, it probably does not matter.
I agree with the previous poster and his concern with melamine. I do think it can be used, but I would use a higher grade product with a more durable pressboard core. But plywood is an even better choice.
I would use a combination of dados, glue and screws at a minimum to hold this thing together. I don't normally do this, but I'd also rabbet a 1/2" thick solid back into the whole structure.
This can be done, I have no doubt about that, but I would suggest rethinking your plan.
For starters, I would consider a cabinet of 48" x 24" x 72" and keeping each level about 18" tall.
Your mentioned dimensions would require 6 full sheets of melamine to complete. The dimensions I suggested would only require 3 full sheets. Do you really want to couble the amout of material needed in order to make your cages 12" deeper front to back which also makes them harder to clean?
If you want to provide more height for climbing and viewing, which I think is a great idea, then limit it to three levels and just build a second unit. This could be done with only 5 full sheets of material. Still one less than what you have in mind.
Think about it.