I'm thinking some sort of malware is at work. Worst case is a keylogger capturing your passwords.
Scanners like malwarebytes can find these things, but safe removal leaving a stable machine is a different proposition.
If it's a bad browser plugin, that's easier. Try deleting all the FF plugins. If that fails, uninstalling then reinstalling FF, taking care not to install goofy plug-ins the second time.
If that fails, scan your PC with malwarebytes. If anything interesting is found, you should seriously consider backup your data, wipe the hard drive and reinstall Windows.