Using 1/2" plywood for the sides, we would need to deduct 1/2" off for each end panel, so your "drop-in" length would be 48" - 1" = 47".
The same thing with the height, we deduct 1/2" top & bottom sheets, so the height of the drop-in back piece would be 24" - 1" = 23".
I've used MinWax stains for most of my own woodworking, but you should be careful to choose a water based stain, so that it will not leach toxic fumes for the next century or so. It still needs to be sealed with something moisture resistant. I have used polyurethane, but there may be some easier to handle and less potentially toxic one can use.
You may already be aware, but plywood comes in varying grades for surface finish, glue type (interior/exterior), material species and treatment for either fire or decay. Most of these variations are described using letter codes.
The better surface grades are "A" (stain grade) or "B" just as smooth, but it is permissible to fill the voids with "footballs" and thus it is considered paint grade. Plywood surface character is denoted by letters for descending grades. You will usually see two letters, one describes the first side, the "face", and the other describes the back. A piece of plywood graded "AA" would have top grade material on both sides. "AB is a more commonly seen type, with a nice outer face and a slightly lesser interior finish. "AC" would be used where the back is not seen at all. This provides a very good surface finish, but a utility grade back that cannot be finished acceptably.
For an animal enclosure, you'll want exterior glue, signified by an "X" in the identifying number. One might see "ACX", which is a cabinet grade face (A), utility grade back (C) and exterior glue (X).
I prefer to work with birch plywood for cabinet grade work, and they refer to it as "luan". It provides a decent stain surface, and can be had in "fancy" grades with nicer grain.
Plywood can also be had with cherry, oak or other hardwood veneer faces that provide very nice stain effects.
HTH,
Ed