The main line (when visible) is designed to always follow the mouse. If you start/stop the line then the line will show up (Start button) or will be hidden (Stop button).

However Line Reader does have "screen shapes" (see the last tab called "Shapes" in the Line Reader interface) with which you should be able to do what you describe here:
"Will I be able to keep the line highlighted on one screen while using the mouse to work on the other"

After selecting (using the check box) at least one shape, please click the "Add to Scene" button.