https://www.youtube.com/watch?v=46MQ1ZMZ-l4

This is a trailer for NBA 2k20, that shows more gambling content than actual gameplay.

The top comment is:

Hey 2k, theres a basketball minigame in your gambling simulator, can you fix it please?…

Could I have a game rendered by a discrete GPU running in a window being handled by an Xorg session rendered by an integrated GPU?

I used to do this, but I use wayland now. Similar setup though.

Would it matter if the video output was physically connected to the discrete GPU or the motherboard, or is that configurable?

Yes it matters. On many laptops the hdmi out is ONLY on the discrete gpu, meaning the discrete gpu must be on (and sucking power) in order to display to external monitors, even if the internal gpu is doing the rendering.