How are the Greek gods usually depicted in Wonder Woman stories? Do they have a constant presence on Earth or do they visit the Earth from another realm? How have they existed for thousands of years without having a larger global impact? Do they hide their existence?
I really don't know anything about them. Where did they come from? What are they capable of?
They exist on Olympus, which i guess is a different realm.
Their importance to the world:
The DCU has many pantheons that exist simultaneously. WW being based on greek mythology mostly deals with the Olympians. Others deal with the Indian gods, and so on.
Their power:
They are manifestations of ideas, much like the New Gods are, if not more.
Characterisation:
Until recently they were played very straight up. Athena is the goddess of wisdom, ergo good, Ares is the god of war, therefore bad. Hermes is the messanger, so he's harmless, and so on. But the truth is that the Greeks saw their gods as capricious and multifaceted beings and themselves at their mercy. Athena was the goddess of wisdom and strategy, sure, but dont say you can weave better than her or she'll turn you into a spider (Arachne). Or dont get raped in her temples (it's your fault, obviously), or she'll turn you into Medusa. Furthermore, Hades, despite being the ruler of the underworld, was a fairly nice guy in comparison to the others. He didnt want to rule the underworld but it's what his brothers left him so he became a ruler - bureaucrat for the dead souls.
Only Azzarello seems to have grasped this concept and his gods are closer to the way they were in Greek mythology. The designs are also very fresh and interesting too.