You can check for recently installed suspicious or unknown extensions.

Start Firefox in Safe Mode to check if one of the extensions ("3-bar" menu button or Tools -> Add-ons -> Extensions) or if hardware acceleration or userChrome.css is causing the problem.

• switch to the DEFAULT theme: "3-bar" menu button or Tools -> Add-ons -> Themes
• do NOT click the "Refresh Firefox" button on the Safe Mode start window

• https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-using-safe-mode
• https://support.mozilla.org/en-US/kb/troubleshoot-extensions-themes-to-fix-problems

Also, at the bottom of the Search panel below the Search Shortcuts box there is a "Restore Default Search Engines" button to restore any built-in search engines that were removed from the Search Shortcuts box. If that is enabled, go ahead and click it and see whether that helps restore more choices to the default search engines popup.