Is there any way to make Thunderbird notify me of new messages without having it in the Unity Launcher. Just like some small notification in the upper bar or like the notification boxes that appear in the upper right corner and disappear after a couple of seconds.
You appear to want to check email (POP3, IMPA4, social networking) without Thunderbird (or similar) being visible (or running).
I suggest you look at an applet like: Popper
Popper – Feature-filled e-Mail notification applet for Ubuntu