Rosetta has been in hibernation mode for most of the last three years. As far as I am aware, "sleep" means using minimum resources since there is nothing for the probe to do during its long transfer orbit. When sleeping, it must still be able to receive commands so that it can power back up.
How does this process work? What electronics on Rosetta are considered crucial and kept running throughout the entire trip?
Q. Was Rosetta completely shut down?
Almost. Only the computer and several heaters remained active. These have been automatically controlled to ensure that the entire satellite doesn't freeze as its orbit took it from 660 million km from the Sun out to 790 million km and back between 2011 and 2014. Everything else on board was shut down, including the transmitters.
The wake-up is controlled by Rosetta's computer. It does not receive an external command, but starts the wake-up sequence based on its internal clock.
The wake-up sequence:
... includes switching on the star trackers, slowing the spacecraft's hibernation spin and switching on and warming up certain systems. Once it has exited from hibernation, the spacecraft will switch itself into 'safe mode' - a standard mode of operation that ensures the basic functionality of the satellite. This, too, comprises several procedures, at the end of which Rosetta will orient itself into Earth-pointing mode and switch the transmitter on.