Hmm, you may want to export your bookmarks and re-download Firefox. When you delete it, also delete all history, options, etc. By doing this, you can see if it is your browser that is doing it. (I guess an easier way may be to use IE or Chrome).
If everything still deletes on your browser, go into your control panel → program (may be different for XP, etc.) and look at the list of your programs. Look into any programs that has to do with clearing history to make computer faster (such as CCleaner) or security systems. You may have the setting on to block all cookies or delete cookies as soon as possible.
However, it is strange that you cannot save on ArmorGames (GCL specifically) as they also automatically save your progress to your account, not just your local computer.