What do these dashed/dotted lines mean in this power cord schematic, and how should I ground this device?
The dotted lines that circle the bundle of signals is often used to delineate a cable bundle (e.g. wrapped together in a sleeve), and when signals are connected to it, that generally means that it represents the (conductive) cable shield. 

The second thing you circled (in J2), is a typical representation of Earth ground. I take it to mean that the chassis is bonded to Earth ground.