Doesn't look like there's a way to single out the Desktop via the 'Manage Storage' GUI.
However, you can always just symlink any directory from your iCloud drive; that way you can ensure that only the Desktop folder is being synced. This is with the caveat that while your Desktop files are being stored in iCloud, they're still taking up space on your HD before, during, and after sync. If you do it on every Mac then you'll keep your Desktop in sync between them all (files added, deleted, changed etc); this might not be what you're after though.
If symlinking isn't something you know how to do, instructions are below.
Backup (via TimeMachine or whatever you use).
Then fire up Terminal and run the following to move your Desktop folder into your iCloud Drive:
mv ~/Desktop /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/
Then just symlink it to the usual 'Desktop' location:
ln -s /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/Desktop /Users/<yourusername>/Desktop
Now anything that you add to ~/Desktop (by dropping things onto your desktop etc) will actually be dropped into your iCloud drive. They are of course still taking up local space on your machine. When you delete something from your desktop then it's gone from the 'Cloud' as well. The benefits of this are more for sync between machines (+ access via your iPhone of course).
If you've got another mac, move anything you value from it's desktop folder into your iCloud Drive 'Desktop' folder first, then delete ~/Desktop on that machine and symlink again as above.
Firstly, the checkbox for Mail just means it syncs Mail.app settings via iCloud and also it enables the iCloud Account in Mail.app or turn it off.
Secondly, System Preferences settings are also synced via iCloud, for example iCloud syncs my preferred Wifi to any new iOS or Mac I enable iCloud on.
Likewise Weather locations for any city you setup is synced over iCloud.
Best Answer
Here's a description of what happens when you turn off Desktop & Documents folder. It might simplify things to move all the stuff that went into the Desktop and Documents folder that was created into the default Desktop folder or Documents folder (or distribute the items between those two folders), then delete the new Desktop and Documents folder to make things the way they were before the option was enabled originally. Then I would suggest logging out and back in before attempting to enable Desktop & Documents folders in iCloud prefpane again.