In Windows 8, when I hit Win+C, a "charms menu" will be displayed to the right. But at the same time, a black square box will be displayed in the bottom-left corner, displaying the current time, WiFi signal strength, a battery icon, etc.
Does this interface item have a separate name that can be googled for available options? Is it accessible with an API in the same way as the charms bar? Is it possible to e.g. add my own icon to it?
It doesn't seem to have an easily located formal name other than referring to it as the Charms clock.
No, there is no published API to modify it or add your own icons to it.
In a Modern UI Windows 8 app (aka Metro), Microsoft would suggest you switch to Live Tiles and Notifications.
This is very interesting case study of converting yet another photo app from iOS to Metro.