Question about "grounded" power supply
No, it is not the way all supplies work. In some cases, just the power supply case is grounded and the output may be floating or have an optional tie to ground (this is the way most bench supplies are wired, excluding some of the big SMPS types). 

However, in this case we can see that the case is made of plastic and there is nothing to ground, so the output is most likely connected to the earth pin. 

There is a third possibility- that earth pin of the supply is only connected to the output through a Y capacitor (for noise reduction purposes), but that is probably less likely.