Would it be possible to make a second fake xorg server such that:

• There is no real physical monitor that it connects to, but the GPU can still do the calculations to render to a fake or "dummy" monitor?
• I can still use another xorg server to display use my actual physical monitors normally
• I can take a screenshot from the fake secondary server and get the picture that would be rendered to a monitor if there was a real monitor being rendered to?
• I can use a NVIDIA graphics card to drive this secondary fake xorg server and the normal one simultaneously?

If so, how would I go about doing this monitorless, simulated graphics environment? I currently most often use GNU/Linux, but I would be open to doing it on Windows, BSD as well if GNU/Linux is not possible.

A nested X server like Xephyr or Xnest?

https://www.dedoimedo.com/computers/xephyr.html

I guess I should have been more specific in my original question, but would I be able to have a mouse cursor inside of the xephyr window doing its own thing (say, controlled by pyautogui https://pyautogui.readthedocs.io/en/latest/) and then simultaneously use the mosuse normally outside of xephyr without disturbing the one inside of xepyhr?

Would this be possible with a qemu virtual machine? I know this would not be possible with a virtualbox one, since the mouse pointer can only be either in or out of the virtual machine in virtualbox, plus I wouldn't be able to use my NVIDIA card in virtualbox because it does not have GPU passthrough to my knowledge.

A alternative might be to create an additional user on your system. You can be logged in normally, but then switch to a second tty, login as the additional user and do what you want in there. You could then switch between the ttys as desired.
If screenshots were saved to a shared directory, e.g. /tmp, they would be available to both users.

The key strokes to switch ttys can vary between distributions. Here, Ctrl-Alt-F1 and Ctrl-Alt-F2 are used and it seems to be the same for Ubuntu.

That sounds like it would work. I'll try this sometime soon. Thanks!

This sounds very much like an X-Y problem: You want to achieve Y, and you think X is the right way to achieve it. You ask a question about X, instead of Y.
So, what are you really trying to achieve? What is your Y?