Have a look in battery saver
It should be obvious what is draining it.
just disable from running in background.
if its facebook grab the new beta it seems fine on mine.
I get 2 days with light intermittent use...
Gaming is ok too in short periods - playing for hrs on end yes it will use lots of power.
Not sure why you are experiencing that but I have experienced no such thing. Battery life on my 1520 is still excellent on 8.1 and no overheating whatsoever
It is wierd but the unit does not heat under 8, im using 8 now and the phone is eextremely cool, but Facebook seems to be huge problem for me or using here drive and gps