that looks amazing!!! I bought some 2x's and 1x's and scrounged every other bit of my bird house from old plywood and other scrap lying around. it's servicable, but nowhere near as nice as what you'll have. Unfortunately i'm not much help on the above questions. I'll toss out my thoughts and you can do as you will?
I used leftover linoleum in mine, on top of some scrounged super-heavy-duty tarping that blew apart off a 'portable garage' unit..... waterproof seemed like a good thing...
I don't know what the blackjack would do in terms of off-gassing? there are some things birds are pretty sensitive to. I think as long as it dries hard and well it should be able to air out and not cause any issues? because that would definitely be better than a tarp. My only other suggestion would be hit up a Mardens or flooring liquidator and see if they have scrap/leftovers of a decent size to put down. Maybe even on top of blackjack, with the idea you could replace it in a few years if it gets worn through?
I love the idea of the deck-with-pool that you can drain to the garden. That sounds brilliant. You will want something not too slippery. maybe leftover hardware cloth on top to keep the flipper feet from sliding around? it will get poopy and very, VERY wet, I would think.
for roofing on my house I have some much-aged (though never previously used!) ripple clear fiberglass sheets. Run roof maybe clear would be nice? you can get sheets of it in either 8 or 12 foot lengths I think. but there are other colors too. it's a sturdy plasticy material that can be used for roofs and even for walls if you wished (like for a sunroom). Otherwise metal roofing maybe? could get hot in the sun...
Your stone/sand/shavings is that for the outside bit of the run? I have heard mixed reviews about sand inside for climates that get really cold, in that the water drains down and freezes and can make for more frostbite. For outside i think there are no objections!
Definitely not an expert here, just tossing out my thoughts. looks like you're doing a beautiful job!