Re cookies.
This computer (running Win7) is set to "accept cookies", 3rd party cookies from "sites visited".
I usually run it with "never accept" 3rd party cookies but changed it to see if it made any difference.
The only plug ins are
OpenH264 Video Codec from Cisco
QVR Host
Silverlight
The last two are not active.
I've tried the site with Firefox on 3 different computers and running under Windows7 and Mint 17.1.
I'm inclined to agree with you regarding cookies or Java but I don't think I've changed anything, other than to try to solve the problem, recently.
Is there a way of checking whether Java is installed/running under Firefox as traditionally, in mint, there was an "icedtea plugin" which doesn't seem to be there now.
That wouldn't affect Firefox under win7 though.
I've got an older machine, which if I can get it running has an older version of Firefox on it - I'll see if that works but may take a while.