Pressing the middle mouse button (or scrollwheel) in many apps allows the user to scroll in the display area by moving the mouse without holding any button.
What is this mode called? What is the name of the icon which appears in the application to indicate this mode is active?

This seems to be referred to as “auto scroll” by various sources. I suspect it is an application level feature as the icons for the 3 major browsers are all different (on Windows 7).