I'm not really sure what you mean by notification icon, I also have several email accounts set up into my S3 and I've put the email app icon (not the widget) on my home screen, whenever I get a new email I get a notification on the drop down menu at the top and on the top left corner of the email app shortcut on the home screen. Just make sure that you haven't disabled the notifications from the quick settings icons in the drop down menu.