I think the desktop printing icon belongs in the system tray. Many times my desktop is full of open windows and having it in the system tray is more convenient for dragging docs to be printed. That is if the system tray supports dnd. The doc can be printed to the default printer if more than one exists. Just a thought.
You can of course drag that icon into the panel, if that's your preference (or 'add to panel' and get it from the menu). I think it might be confusing though if there are two printer icons on your panel during printing something though. (When a print job is being processed, a printer icon appears in the system tray.)
I did not know an icon shows up in the system tray while printing. How about having a normal icon in the system tray permanently and while printing it changes into an animated icon of some sort?
We'll move the Print icon to the panel as a launcher.