I sure wouldn't want to try cutting the glass myself, but fyi, you can usually get a pro to do it for not too much money- at least in the small town where I live you can, so I figure that's a fairly good prediction for elsewhere. Contact the local glass & mirror installers- I've had them cut some glass for a picture frame before, when mine got broken. And I also had them cut a piece of glass to divide a glass tank, & was considering other projects as well. In fact, if you tell them what you need, they might even have the scraps to supply you? Never hurts to ask- & they might prefer working on glass they can vouch for.
I agree with Malum- I'd want 1/4" glass too. And also, wood can be sealed so it's waterproof, if you use the right stuff (just make sure it doesn't off-gas & harm your animals- many products are not pet-safe.) As far as lining a wood enclosure with glass, it sounds like a lot of work but I don't see why it wouldn't work, & it would really insulate the "tank" too. The main thing is making sure all the seams are sealed thoroughly with aquarium (silicone) sealant- & you don't want it too thin (for durability) but if you lay it on too thick, it takes forever* to dry because the interior stays moist while the outside slowly dries. You must ensure it has stopped off-gassing fumes before having critters move in. It's smells like vinegar & is very irritating to their lungs. (*If it's too thick, it might take weeks.)
I've done some creative enclosures out of furniture cabinets in the past, but never tried this specifically. Hope it works out for you.