Why does everything on my Desktop show up on all my desktops? I have multiple desktops defined, which I use for different projects. How come everything I save to my Desktop when I'm working in, say,  Desktop 1 shows up on all my desktops?

Maybe I don't properly understand Desktops.
What you are referring to are called Spaces in macOS lingo.

While Spaces are named as numbered desktop as Desktop 1, Desktop 2 etc. they are essentially all the same, except giving you are a means of organizing app windows.

Each Space gets access to the same Desktop and hence shows the files available in it. 

If you choose to use external display(s), the Desktop is not shown on the Spaces of the non-primary displays.