I'm Running Xubuntu 16.04 LTS 64-bit, but this is a general question.
In Power Manager, there's a checkbox for "Handle display power management". What handles display power management when this box is unchecked?
Additionally, what determines how the system behaves if I uncheck Power Manager from the Startup list? What determines what happens if the power or sleep buttons are pressed, if the battery is low, or if the system is idle for a long amount of time?
Xubuntu uses XFCE desktop environment as default. So the power manger is more likely a xfce application which is called xfce4-power. You can read the official documentation of the specific preference for this app in here: http://docs.xfce.org/xfce/xfce4-power-manager/preferences#display
If you checked "Handle display power management" the box, you could deal with below three display options.