Having the Java Plugin icon in the system tray is normal behavior - I don't think having an internet explorer task up in the task list is normal, but I'm not an expert at how IE is handled as a process across all versions of Windows. I'd
test it out by opening some web pages that don't contain your applet and see if you see the same behavior.
I would ask if you are using any threads in your applet, but I don't see how this could keep the JVM in the browser alive after IE has been completely exited.
Also, this question is in the wrong forum, I will move it to the Applet forum, where you may get more help with this...