Is earth of a 3-prong AC/DC adapter is connected to the GND of DC output?
Ground is quite often a label for "zero volts" or reference or common. As opposed to "earth" which has a more particular meaning.

Generally speaking our circuit ground is not connected to earth and personally I prefer to use different symbols to emphasise that.

Unless there was a specific requirement for circuit ground to be connected to earth, I would never connect them together. The DC output is then said to be floating (the negative/ground output doesn't have a reference to some other circuit elsewhere on your bench). The main reason to do this is to avoid ground loops that could occur if you connect your microcontroller (say) to another piece of equipment (computer, etc).