Baltic Birch does not stain very well. It gets kind of blotchy. If the pieces you’re using in your end product are small enough, that doesn’t matter. (i.e., I’ve made earrings out of it, and they’re tiny, so the blotchiness doesn’t show up) Also, I have stained/dyed it black (Minwax True Black), and that works as well, since it is all dark.
It paints fine, I’ve used both brush on and spray paint with Baltic Birch.
On the plus side, I can get BB 1/8" plywood for less than $0.50/square foot. So it’s very cost effective for me. And I mostly use clear coat finishes, like polyurethane, so the staining issue isn’t an issue.