[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?
You can generally back up your home folder and use it with another distro, without losing data. (though it can happen that stored settings are not compatible with some changed programs).
All your data is in your home folder.