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

You may be in Private Browsing mode or told Firefox not to remember.

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

Turn off Always Use Private Browsing Mode Turn on Remember My Browsing And Download History 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.

Macs: Make sure you install Firefox properly and that you do not run Firefox from within the DMG (Disk Images) file. https://support.mozilla.org/en-US/kb/how-download-and-install-firefox-mac

The credentials to login automatically to the Sync account are stored in the Password Manager (logins.json in the profile folder; chrome://FirefoxAccounts), so make sure you keep this file because otherwise you need to login again.

If you use external cleanup software like CCleaner then check the settings for Firefox.

You appear to have a user agent that is modified by Avast Anti-Track that could be causing problems, so if you do use this software then try to disable it. You will have to contact Avast about this if their software causes problems for you. We can't do much about this apart from identifying Avast Anti-Track as a possible cause.