How do you Airplay something on Macos Monterrey? On a multi monitor setup, why is it impossible to drag a window over to a monitor where another window is full

How do you Airplay something on Macos Monterrey? On a multi monitor setup, why is it impossible to drag a window over to a monitor where another window is fullscreened? How do you select a specific resolution or refresh rate when you're plugged into an older meeting room projector?