There is software like Advanced SystemCare with its Surfing Protection feature that can protect files in the Firefox profile folder against changes. If you have such software then check the settings or uninstall this software.

If you do not keep changes after a restart or otherwise have problems with preferences, see:

• http://kb.mozillazine.org/Preferences_not_saved

See also:

• https://support.mozilla.org/kb/remove-toolbar-has-taken-over-your-firefox-search-

Do a malware check with several malware scanning programs on the Windows computer. Please scan with all programs because each program detects different malware. All these programs have free versions.

Make sure that you update each program to get the latest version of their databases before doing a scan.

• Malwarebytes' Anti-Malware:
  http://www.malwarebytes.org/mbam.php
• AdwCleaner:
  http://www.bleepingcomputer.com/download/adwcleaner/
  http://www.softpedia.com/get/Antivirus/Removal-Tools/AdwCleaner.shtml
• SuperAntispyware:
  http://www.superantispyware.com/
• Microsoft Safety Scanner:
  http://www.microsoft.com/security/scanner/en-us/default.aspx
• Windows Defender:
  http://windows.microsoft.com/en-us/windows/using-defender
• Spybot Search & Destroy:
  http://www.safer-networking.org/en/index.html
• Kasperky Free Security Scan:
  http://www.kaspersky.com/security-scan

You can also do a check for a rootkit infection with TDSSKiller.

• Anti-rootkit utility TDSSKiller:
  http://support.kaspersky.com/5350?el=88446

See also:

• "Spyware on Windows": http://kb.mozillazine.org/Popups_not_blocked

The weird thing is that it when I click the down-pointing arrow to select search engines, it doesn't list bing as one of them (because I removed it), but it still resets it to bing every time I reopen firefox

Hi drtricky, I don't know whether you can actually delete built-in search plugins completely, but Firefox should remember the last search engine you used in your previous session and start up defaulting to that one in your next session.

Unless the main settings file won't save correctly, or it is overridden by an external settings file or external software or an extension. Please see cor-el's reply for details on further diagnosis.

Deleting built-in search engines would involve to remove its XML file from the browser\searchplugins folder in the Firefox program folder, but such changes are not guaranteed to survive an update of Firefox.

Custom installed search engines are installed in the searchplugins folder in the Firefox profile folder.