first of all... windows 2000... (*shiver), it is ok.... it's better than ME, but that's about all the praise i can give it without getting sick.
Your OS aside, the coffee cup symbol that you see is for applets and other items that require the JVM in specific contexts, but most applications which happen to use java, can access its virtual machine without the added wait time of the icon in the system tray.
my opinion would be to remove both of them, complete uninstall of limewire, Java, JSDK, and depending on the number of programs possibly remove any other java intensive programs you have. once this is done, start by installing the current version of java, looking at anywhere from 30min-60min for a full installation. Once this is done all registry keys are properly in place for other software, then install your limewire once again, and test it. If it is working continue installing any other programs you may have uninstalled.
If it does not work, check the settings of your firewall or anti-virus. Another odd check as to whether it is a system setup error is to create a new user account and repeat the process for the new user and see if it works.