Using Dropbox Folder as Desktop
Though I have not actually done this it should be possible to redirect your desktop folder to a different location. Since the Dropbox folder is just a folder in your user profile (that the Dropbox app then knows what to do with).

I would avoid using the dropbox folder itself as the desktop folder just to avoid any difficulties or conflicts between macOS and the Dropbox app. Rather I would put a folder called "Desktop" in the Dropbox folder and create a symlink in place of your ~/[username]/Desktop folder that points to ~/[username]/Dropbox/Desktop.

If you are not familiar with symlinks (the Unix version of an Alias but works with system processes) this link from GigaOm should be helpful.

The only question that remains is will macOS allow you to do this as the default folder structure in your user folder may be protected by the recently introduced "SIP" or System Integrity Protection system.