The address bar of my Firefox is not suggesting any previously visited sites under the drop-down menu.
Check Options > Privacy, under "Location Bar". Have you selected History? If Nothing is the selection showing, you will not get anything in the dropdown list.
You can also check the value of the browser.urlbar.maxRichResults pref on the about:config page.
You can also start Firefox in Safe Mode to check if one of the extensions or if hardware acceleration is causing the problem (switch to the DEFAULT theme: Firefox/Tools > Add-ons > Appearance/Themes).