I use plywood for cages all the time. Use a quality AB or AC grade ply for exterior use. To seal the cage initially, I use a tung oil product. Then I follow with a natural tinted paint or a stain. Top coat is clear polyurethane for two to three coats. Nothing will get into the wood then. The cages must be aired out. One other detail is to caulk all seams and cracks with an aquarium grade silicon. This removes place for mites and chiggers to hide.
You can check wiith
Things to Use for a more complete discussion of paint and sealer types. Be aware that even child safe products have some toxic chemicals.
There are some fiberglass sheeting materials that come in 4 x 8 sheets that can be attached to the wood with simple fasterners and/or glue which would work well. Nice easy to clean white surface. You see this in public bath rooms used a splash backs.
Good luck - again there are no wrong choices - just choices that work for you!!!