How does the power efficiency of an appliance compare using 12V DC or 240V AC with an inverter?
First question regarding the power consumption of some arbitrary device that accepts 12VDC input or 240VAC input cannot be known without making measurements of the current draw from the source and making the necessary calculations. There is no way to know how a black box device converts either input to its required operating point and how the efficiency of the methods will vary.

The use of an external inverter to convert 12VDC to 240VAC will not be a 100% efficient process. Once again it would not be possible to really know the inverter efficiency without measurements and calculations. 

Once you knew the efficiency of the inverter you could factor that into the differences of power usage by the device that you have measured. But it can be said that if the device itself consumes less total power when operating at 240VAC versus when operating at 12VDC then there may be the possibility that using the external inverter could be more efficient overall.