I was baffled by the question " How do program settings work in virtual desktops?" because it never occurred to me that people would consider virtual desktops to be some sort of secure program separation.
It's not doing anything of the sort. (After all, it says so right in the name: virtual desktops.)
Virtual desktops are purely a window management feature. They help you organize your windows. Asking whether windows on different virtual desktops have isolated settings is like asking whether windows on the left hand side of the screen have isolated settings from those on the right hand side of the screen.
If the virtual desktop feature had been built out of actual window manager desktop objects, you would be pretty unhappy. Windows in separate window manager desktops cannot interact with each other, so you couldn't drag from one window to another, or copy/paste between them. A window cannot move between window manager desktops, so you would be stuck with whatever desktop the window originally appeared in. And features like "Show on all desktops" would make no sense at all.
Instead, when you put a window on a virtual desktop, all you're doing is giving the system instructions on when you want to see the window and when you don't. The system shows the window when that virtual desktop is active and hides it when the virtual desktop is not active. If you say "Show on all virtual desktops", then the window is always shown. Moving a window between virtual desktops is just changing the rules for when the system hides and shows the window.
Dirty secret: I always forget that virtual desktops exist. I just keep everything on one virtual desktop.