Unless something has changed very recently, Destiny 2 does not work under Proton because the Anti-cheat isn't compatible. Windows is an option, but I'd really recommend a dedicated Windows handheld for that instead of the Deck.
I haven't tried Windows on the Deck itself, but based on 6800u performance, you're looking at maybe getting a solid 30fps on lowest settings. Destiny 2 is not a very well optimized game atm, and even the Ally at 20w can have trouble holding good performance in Neomuna or the Tower.
Maybe, but unless we're in there, it's hard to say for sure. Battleye supports it but given the clusterfuck that is the Tiger engine, they might be using an old or heavily modified version. Especially given how easily cheaters are getting past it atm, I would not assume they're keeping up to date on it without some evidence.
Stadia limits the player's control of the system substantially and might not have even needed the anticheat, or the same anticheat, given it's closer to a console in terms of user's ability to run programs alongside the game to modify memory.
The Deck is designed for SteamOS and SteamOS is pretty much designed for the Deck. They're chocolate and peanut butter.
I feel Valve's Windows support is not great -- it's a custom APU for them so you're getting your drivers through Valve, not directly from AMD's Adrenalin. The APU drivers are from March, I'm not sure if we've seen any Linux improvements since then on the APU side but months without GPU driver updates to address game compat issues doesn't encourage me.
They don't officially support dual booting AFAICT, and given the size of modern games + Windows, I wouldn't want to dual boot on any of the offered SSDs tbh.
If you've got a deck and really want a Windows-only game, it works. But given the easy availability of the Ally, and the upcoming non-extreme Ally, I can't imagine recommending anyone who wants to primarily play a Windows-only game get a Deck.
I've done it so it's definitely possible, though I can't remember if I used Steam link or Moonlight to do it. It was also pre-Lightfall, so they definitely could've changed it.
estiny has some anti-overlay stuff enabled that limits what can pop up on top of it, which might also be affecting some form of streaming?