There might be some malware or an extension that was added to Firefox, I do not know exactly what caused this, but I can recommend malware scanners as well to see if something is detected that current antivirus or firewalls did not capture: Sometimes a problem with Firefox may be a result of malware installed on your computer, that you may not be aware of.

You can try these free programs to scan for malware, which work with your existing antivirus software:

• Microsoft Safety Scanner
• MalwareBytes' Anti-Malware
• Anti-Rootkit Utility - TDSSKiller
• AdwCleaner (for more info, see this alternate AdwCleaner download page)
• Hitman Pro
• ESET Online Scanner

Microsoft Security Essentials is a good permanent antivirus for Windows 7/Vista/XP if you don't already have one. Windows 8 has antivirus built-in already.

Further information can be found in the Troubleshoot Firefox issues caused by malware article.

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I don't know, but I have a guess.

Firefox 43 includes enhanced support for virtual keyboards on Windows tablet and convertible devices. Part of this feature includes checking for a physical keyboard before displaying an on-screen keyboard. Could that be the problem??

For testing purposes, you could try this:

(1) In a new tab, type or paste about:config in the address bar and press Enter/Return. Click the button promising to be careful.

(2) In the search box above the list, type or paste osk and pause while the list is filtered

(3) Double-click the ui.osk.detect_physical_keyboard preference to switch its value from true to false

Does that make any difference either during the rest of your session or at the next startup?

And/or if your keyboard isn't detachable, you could completely disable the on-screen keyboard feature:

(4) Double-click the ui.osk.enabled preference to switch its value from true to false