Cheaper plywoods can be made to work; you just might end up with some mini-projects mid-project.
My first closet project used general finishes wipe-on gel urethane. You don't need many coats (I might have done 2 or 3). The biggest question with oil-based varnishes is clearing the ambient dust before applying, and letting it harden enough before a light scuff sanding. I didn't use any stain.
I did a second project with a danish oil finish. This was even easier to apply, but took much longer to settle down to a state where I'd put clothes anywhere near them.
Your joints look good.
I'd recommend keeping the joint in clamps until the glue sets up. Few things as lousy as finding a joint slightly opened up and you now have a trapezoid you're trying to use with drawer slides...
Edge banding took some work to get trimmed nicely. It is probably faster than solid edging, but I just didn't like the process as much (not as confident working with the glued edges compared to solid wood)
Sounds like this will be a great project!
Matt