When I was doing woodworking in 4-H as a kid the guy who was teaching me said that a good woodworker isn't one that doesn't make mistakes but one that knows how to cover them up so according to him you're doing it right.
RE: face frame - Prepping for painting(and painting itself) is an entire art in itself, really. I'm not very good at it, but you really have to take your time, use a filler/epoxy on any even remotely small holes, but especially near the seams. Sanded from 120 - 240, then wet it several times to raise grain and lightly hand sanded those each time. Then lightly sanding between every coat of primer(3) and every coat of paint(3). I won't be making a lot of painted pieces like this, but the wife wanted it blue.
Walnut just oxidizes and fades with age unless you protect it with a finish. Sunlight, heat, air, all accelerate. How old is that stack?
I both like it and don't like it at the same time. It's a neat idea but those axe toes sticking up, even if they have no edge, would get annoying.
I wouldn't complain about 2ftx2ft sheets. Only thing I have local is home Depot and they don't have shit. Cabinets shops don't have anything over 0.75in.
Have you experimented with gluing up some Baltic Birch yourself? The demand for 1 inch+ ply is probably so low that it has to be silly expensive.