Start Firefox in Diagnose Firefox issues using Troubleshoot 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).

• Don't make any changes on the Safe mode start window.
• https://support.mozilla.com/kb/Safe+Mode

The website may be using autocomplete=off to prevent Firefox from saving the name and password.

You can remove autocomplete=off with a bookmarklet to make Firefox save the name and password.

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

• Saved Password Editor: https://addons.mozilla.org/firefox/addon/saved-password-editor/

I tried everything, too, including deleting the default profile and creating a new one, re-installing FF after cleaning the Windows registry and deleting all Mozilla files&folders, etc. I found out that our problem is related to nVidia graphic cards driver and Firefox`s use of hardware acceleration. There are 2 solutions that worked for me in Windows 7: 1.) go to Tools-Options-Advanced and uncheck "Use hardware acceleration when available", if you can live with reading crappy fonts on websites or 2.) right click on the desktop-Personalize, then change your theme to a non-Aero one. Mine is Aero 7 Basic. Everything works like a charm now.

Thanks for solution zedog :)