In Windows 95/98, open the Dial-Up Networking Folder under My Computer.
Click on "Connections", from the Dial-Up Networking toolbar, then UNcheck "Show an icon on taskbar after connected".
As far as why the icon causes stutter,..well, the puppy uses quite a bit of CPU cycles to do the animation and continues to run, even when it is not visible.
I have no clue about Zone-Alarm as I do not use any personal firewalls.