Try the Firefox SafeMode.
A troubleshooting mode, which disables most Add-ons.
(If you're not using it, switch to the Default Theme.)

1. You can open the Firefox 4/5/6/7 SafeMode by holding the Shft key when you use the Firefox desktop or Start menu shortcut.
2. Or use the Help menu item, click on Restart with Add-ons Disabled... while Firefox is running.
   

To exit the Firefox Safe Mode, just close Firefox and wait a few seconds before using the Firefox shortcut (without the Shft key) to open it again.

If it is good in the Firefox SafeMode, your problem is probably caused by an extension, and you need to figure out which one.
http://support.mozilla.com/en-US/kb/troubleshooting+extensions+and+themes

When opening in SafeMode with addons disabled, the search bar suggestions worked. How do I identify the source of the problem now?

Thanks!

Did you try this?
https://support.mozilla.com/en-US/kb/troubleshooting%20extensions%20and%20themes#w_disable-all-extensions

I have disabled all addons and restarted Firefox normally (not safemode) - the suggestions do not work....

Try a new Profile.
http://kb.mozillazine.org/Creating_a_new_Firefox_profile_on_Windows

Starting FF with another profile - search bar suggestions work. So what's the next step? I don't want to start fresh with a new profile...

Short of wasting a lot of time trying to figure out exactly what's wrong with the old Profile, my advice is to stick with the new Profile and recover your personal data files from the old Profile.

Copy specific files from the old Profile to the new Profile.
http://support.mozilla.com/en-US/kb/Recovering+important+data+from+an+old+profile

I'll give it a try - thanks!

You can try to reset the prefs in your old profile as you do seem to changed a lot of them.

Try "Reset all user preferences to Firefox defaults" on the Safe mode start window.

• https://support.mozilla.com/kb/Safe+Mode
• http://kb.mozillazine.org/Resetting_preferences

I found the problem - it was the search plugins. I deleted the content of the "searchplugins" folder (but not the folder) and the search suggestions are working again!