Several issues. Let's start here:

Updating Firefox

Is the method in this article not working: Update Firefox to the latest release?

Thank you for sharing your browser details. It appears you get your Firefox from your repository since you have a Ubuntu extension. I don't know why your package manager isn't getting the latest version, but perhaps there are some dependencies that need to be installed first? You would need to check with them.

If you want to try installing directly from Mozilla, you can download from here: https://www.mozilla.org/firefox/all/

I do not know the way to switch from the Ubuntu distribution to the Mozilla one; you may be able to find some tips and tricks on that here or on Ubuntu forums.

Signing into Gmail

Does Google accuse you of entering the wrong password, or does it just keep cycling back to the login page?

Could you try:

New Profile Test

This takes about 3 minutes, plus the time to test your problem site(s).

Inside Firefox, type or paste about:profiles in the address bar and press Enter/Return to load it.

Click the Create a New Profile button, then click Next. Assign a name like Dec2018, ignore the option to relocate the profile folder, and click the Finish button.

After creating the profile, scroll down to it and click the Set as default profile button below that profile, then scroll back up and click the Restart normally button. (There are some other buttons, but please ignore them.)

Firefox should exit and then start up using the new profile, which will just look brand new. Please ignore any tabs enticing you to connect to a Sync account or to activate extensions found on your system so we can get a clean test.

Does Gmail allow you to log in in the new profile?

When you are done with the experiment, open the about:profiles page again, click the Set as default profile button for your normal profile, then click the Restart normally button to get back to it.

This may or may not have anything to do with your (current) problem, but your system details show that you have this extension installed :

Email 4.0.17 (@Email)

It's malware - would you please remove it ?