BibDesk appears to be in the Dropbox folder and changes in settings are not saved

Thanks, @Mark.

1. You write, "Dropbox only does this if you actively set up/activate the BackUp feature on a machine." I just looked at the Backups tab in Preferences. When I select Manage backups, I see the option to "protect more files with Dropbox Backup", which three checkboxes for Desktop, Documents, & Downloads. None of them is checked. If I'm using the feature (and indeed I see those three folders in my Dropbox folder), why are they not checked, and how can I fix the problem?
2. Now I looked at the sync tab. So it does seem like I chose to select many "folders to sync to this Mac". Apparently, when I set this up, I hadn't noticed that among them, I had selected the folders, Mac and Mackup (I don't remember having a folder called Mackup. Do you know what it could be?)
3. Having unchecked these two, the Dropbox folder does have either a Desktop, Documents, or Downloads folder. But when I look at ~/Library/Application Support, BibDesk is still connected to Dropbox. I'm going to restart my machine, and see what happens.
4. In your opinion, could this be the cause of the issue I'm experiencing, i.e., open-source software (e.g., Path Finder, BibDesk) not retaining settings between launches? I have a test user set up where this is not happening, so it must be something peculiar to my set-up, and Dropbox is my top suspect.