Okay, so. Just as I feared, you can't really delete Firefox from Linux Mint. You can remove it via the package manager, but all the base files are still there, and the system doesn't give an option on the right-click menu to delete them. BUT.
After digging around for a bit, I found an alternate solution. Firefox, at least in Linux Mint, offers an option (well buried on the troubleshooting information page, triple bar menu->Help->Troubleshooting Information) called "Refresh Firefox." Much like uninstalling and clearing out base files in Windows does, this restores Firefox to default, pristine, just-like-it-just-installed state. And then after I did that, I power-cycled my laptop, then reloaded all my add-ons, bookmarks and open tabs. And that seems to have fixed it. Gmail, tumblr and youtube all seem to be working properly again, and I'm typing this out in firefox.
Caution for anyone else doing this: BACKUP YOUR DATA. Write down your extensions and settings, if you're using stylus (like I am) export your styles, if you're using New XKit (like I am), use XCloud to export your extensions and settings, and backup your bookmarks and open tabs. (To export your open tabs, right click on a tab, click "select all tabs" then right-click again and click "Bookmark tabs." I create a folder with the current date so I know what's what. Then backup your bookmarks. Then when you have Firefox refreshed and restarted, import your bookmarks, then open the bookmarks menu, find that folder, right-click on it and choose "open all in tabs."
Yes, if you use sync you can do this much easier, but I don't trust cloud storage/cloud computing, and I'd rather have everything under my physical control. Plus if you're using sync and the problem persists even after a refresh, then try this method. It works in Windows, too - my old laptop was running Windows 7, I switched to LInux after I got my first look at Windows 10 on my new laptop, and this is how I migrated everything.
I'm gonna go ahead and mark this as SOLVED.