Cool, so many different browsers doing different things so I can only generally suggest things to try.
One BIG thing that may be causing problems for some of you is that Windows XP is no longer being supported as of 8th April so any updates after that date to any of the browsers and other essentials for web browsing such as Java are no longer checked to see if they run properly in XP!
There will be compatibility issues eventually, much like if I'm messing about in Windows 95 or 98 for fun(I'm a sad geek) I can't surf the web using a modern browser!
I'm not sure how tech minded you are so ignore anything you have already tried, here's a few things;
Firstly, run a good system cleaner of some description, Piriform CCleaner is what I use, then;
Try logging in with another browser, having clicked remember me, leave it logged in and close the browser, open it again and see if you get the same problem.
If you don't then it's likely corrupted older(friendly) cookies in Firefox causing the problem and you can clear the cache/cookies manually from

making sure to select "Everything" in the "Time Range to Clear".
If you still don't get any joy open

again and go check permissions for individual websites in the tools section.
Other than that I'll try and help if you still don't get any joy but that's a whole new can of worms getting an old OS to behave with a browser designed for Windows7/8!