If you choose to have new mail messages notified by an icon in the system tray, then the icon should disappear if you switch to, or refresh your inbox.
The indicator is fairly useless unless you are in the habit of double-clicking it to open new mail.
I would ...
Have a visual live time indicator on calendar views.
When calendar is quite busy a time indicator would help to focus on current/upcoming appointments/schedules. The indicator would show up in the current day at the current time.