[Edit] To answer my question, yes, Dropbox does indeed store its application information in your user’s /home folder by default. As long as you don’t wipe your /home folder, you should be good to go once you reinstall the Dropbox app after reformatting/reinstalling your distro (Tested with a few Fedora-based distros, YMMV if you use Debian/Arch). I didn’t have to re-login; the Dropbox app just worked.
Is it possible to reinstall Linux (or distro hop) without losing my Dropbox install? Could I move the Dropbox install to my home folder so it survives the OS install?
It is of course possible but you’d likely be causing a big mess on the filesystem. If you’re able to move the install into a home directory, why not just archive what you need and restore it after reinstall? This would be the cleaner way to go about it.