I know it's for Steam Deck, what I'm interested in knowing is whether the games work with Aurora only in desktop mode or also in big picture mode
Oh, you meant gaming mode as Big Picture Mode is a completely different thing. Yes, it will. But not 100% of the time. It does pretty much work 99% of the time in desktop mode.
So I downloaded and setup the tool for the first time. When i open it say aurora and game status stopped. Did I do something wrong or miss a step?
If you launch the Steamdeck Tool launcher by itself it should say stopped for both. It will only change when you launch a game.