Let’s do a full clean re-install;

Note: Firefox comes in three or more folders on all computers. They are;

Maintenance: (Programs Folder) <Windows Only> Firefox itself: (Programs Folder) And two folders in the profile of each user on the computer for each Firefox profile for that user.

If you remove the Firefox folder, the user profiles would not be affected.

Download Firefox For All languages And Systems {web link}

Save the file. Then Close Firefox.

Using your file browser, open the Programs Folder on your computer.

Windows: C:\Program Files C:\Program Files (x86) Note: Check Both Folders

Mac: Open the "Applications" folder. https://support.mozilla.org/en-US/kb/how-download-and-install-firefox-mac

Linux: Check your user manual. If you installed Firefox with the distro-based package manager, you should use the same way to uninstall it. See Install Firefox on Linux; https://support.mozilla.org/en-US/kb/install-firefox-linux

If you downloaded and installed the binary package from the Firefox download page, simply remove the folder Firefox in your home directory. http://www.mozilla.org/firefox#desktop ++++++++++++++++++++++++++++ Look for, and remove any Mozilla or Firefox program folders. Do not remove the Mozilla Thunderbird folder if there is one.

Do Not remove any profile folders.

After rebooting the computer, run a registry scanner if you have one. Then run the installer. +++++++++++++++++++++++++++ If there is a problem, start your Computer in safe mode and try again.

http://encyclopedia2.thefreedictionary.com/Linux+Safe+Mode This shows how to Start all Computers in Safe Mode; Free Online Encyclopedia

So, yes AND no. Once I ran software to kill the broken links, it allowed me to load a copy of firefox.

However...

It crashed. Then I started getting the same message back: Firefox is already running.

When I rebooted, there was a new folder on my desktop called "Old Firefox Data".

When I started Firefox after the reboot, it brought it up but then will not allow me to log in (hit the login button and...nothing.

Haven't done it through Safe mode yet, but so far, this is really screwed up.

There are many reasons why a web page may not load successfully in your browser. Sometimes the problem is one of compatibility, such as occurs when a website’s developers use proprietary coding techniques that not every browser knows how to interpret. You can check for this type of issue by using a different browser to visit the website in question. That’s one of the reasons why it's a good idea to keep Safari, Firefox, and Chrome web browsers handy. If a page loads in one browser but not another, you know it’s a compatibility problem.

One of the most likely causes of a web page not loading is a wrongly configured or poorly maintained DNS (Domain Name Server) system by your ISP (Internet Service Provider). Most internet users have the DNS system assigned to them by their ISP. Sometimes, this is done automatically; sometimes, an ISP gives you the DNS server’s internet address to manually enter into your Mac’s network settings. In either case, the problem is usually at the ISP’s end of the connection.

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Go to that message, and use the Undo link to its right.

Dunehunter said

When I rebooted, there was a new folder on my desktop called "Old Firefox Data".

Sometimes a problem causes Firefox to reset itself. The original profile is stored in the "Old Firefox Data" folder on the desktop while elements are copied to a new profile.

Refreshing Firefox {web link} is a LAST RESORT ! !

Refreshing Firefox -- easily fix most problems

If you're having problems with Firefox, refreshing it can help. The refresh feature fixes many issues by restoring Firefox to its factory default state while saving your essential information like bookmarks and open tabs. Everything else, like user settings and add-ons, are removed. Note: Refresh saves your old profile to the desktop in a folder called Old Firefox.

Since I can't get to Firefox, I can't refresh it. As you say it's a "last resort", this suggests that I will have to take the computer back to a previous state.

If you have problems accessing the Profile Manager, you can create a new default Firefox profile by deleting the profiles.ini file, using these steps:

   1.Click on the Windows Start button and click on Run...
   2.Type %appdata% in the Run box and click OK. ...
   3.Double-click the Mozilla folder.
   4.Double-click the Firefox folder.
   5.Delete (or rename, for example, to profiles.iniOLD) the profiles.ini file. 

https://support.mozilla.org/en-US/kb/how-run-firefox-when-profile-missing-inaccessible