Type about:preferences#security<enter> in the address bar. Look under Logins,

Note: For Firefox v56+ its; about:preferences#privacy Look under Forms & Passwords +++++++++++++++++++++++++++++++ Turn on Remember Logins. To check the passwords you have listed, press Saved Logins.

Press the Exceptions button and see what's inside.

Warning: There are some web pages that do not allow browsers to record or fill in the log-in information.

But if you go to the Mozilla Add-ons Web Page {web Link} (There’s a lot of good stuff here) you can try a Form Filling add-on. I found that the form-filler add-on I use is not affected.

Your ‘logged-in’ status is stored in special cookies. If the cookie is removed, you are logged out.

Type about:preferences#privacy<enter> in the address bar. The button next to History, select Use Custom Settings.

At the bottom of the page, turn on Clear History When Firefox Closes. At the far right, press the Settings button. Turn on ONLY Cache and Form And Search History leaving the others off.

In case you use "Clear history when Firefox closes" or otherwise clear history.

• do not clear the Cookies
• do not clear the Site Preferences

• Options/Preferences -> Privacy & Security
  Firefox will: "Use custom settings for history":
  [X] "Clear history when Firefox closes" -> Settings
• https://support.mozilla.org/en-US/kb/delete-browsing-search-download-history-firefox

• clearing "Site Preferences" clears exceptions for cookies, images, pop-up windows, and software installation and exception for password and other website specific data
• clearing "Cookies" will remove all selected cookies including cookies with an Allow exception you may want to keep

Let the cookies expire when Firefox is closed to make them session cookies instead of using "Clear history when Firefox closes" to clear the cookies.

• Options/Preferences -> Privacy & Security
  Cookies and Site Data: [ ] "Delete cookies and site data when Firefox is closed"

You can create an Allow exception to keep specific cookies.