Why does the phone light up at times but other times it's dark with a green dot? Why does the phone light up at times () and others it just black with a green dot next to it?
The grey dot and a device shown with the display off is used when the device is offline and cannot be tracked with Find My iPhone.
The green dot and a device shown with the display on is when the device is actively reporting its location to Find My iPhone.


A green dot and an 'off' display as you mention in your question is mutually exclusive.