Can a god die in Greek Mythology?

No, they can't. THey are usually confined to Tartarus when they need to be got rid of, but they are still alive there, only asleep and not really enjoyong themselves. The only example I can think of where a full immortal dies is Chiron the centaur. He was the sno of Zues, adn was half man half horse. He was poisoned by Heracles' arrows, by mistake, which had been tipped with the blood of the hydra. THis caused a wound that could not be healed. Chiron faced an eternity with this pain, becasue of his immortaliity. He asked Zeus to let him die, asking him to set Prometheus free in place of his life. Zeus did as he asked, and Chiron was placed in the sky as a constellation.

There are no other examples of immortals dying, in Greek mythology, at any rate.

Yes Gods can die. It can happen in several ways... in Greek Mythology there are several examples. All the Olympians are descended from the Titans. And the Titans are descend from the Primordial Titans/Elder Greek Gods. The first recorded death is Ouranos, who is sent into living death by his son Kronos. Although he did not "expire" he no longer interacted with the world and just ceased to function as a being...and becoming the sky. Kronos on the other hand fearful of retribution at his own children. Although they weren't be killed out right they were slowly being absorbed. This is a version of death, total absorption into another being. It takes a very long time, but a gods power and personality is taken into a greater being and soon the individual deity is no more. Kronos however never finished the job. Immortals can get hurt...Prometheus was a Titan himself.... and each day he is made to suffer up until the point of death. Again, Kronos castrated his father (injury) he could not recover from. Zues did the same thing to Kronos. In an ending myth with Zues and the 2nd uprising of the Olympians... he himseld snapped Hera's neck. Also what was odd if these being truly where immortal why were the dependent upon Ambrosia or Nectar to retain their immortality? It is proven that a gods physical shell can be destroyed, but spirit part can be absorbed, made dormant or dispersed.

I don't think they can die in the traditional sense, but that they "fade away." I recall reading a story about the death of the Greek Gods, who faded away into non-existence when they were no longer worshipped. They were replaced by Christianity, and they found that without burnt offering, altars, and prayers, they simply ceased to be. Of course, this is a mythology about a mythology, so it's not really a Greek myth. I can't recall exactly what book I read it in.

As an aside, death was an actual God called Thanatos, who was mainly associated with peaceful death only. Violent death was called Keres. As personifications of Death, would they themselves be considered dead? That's really a philosophical question.

As for the series Hercules, they simply added some additional drama to the story of the Ceryneian Hind and the Third Labor of Hercules. It has not such power to kill Gods, otherwise it would not have been sacred to Artemis. See http://en.wikipedia.org/wiki/Ceryneian_Hind .

No gods in Greek mythology are called "athanatos" = undying, unkillable.

Some examples.

When Typhon defeats Zeus, he cannot kill the god so cuts out the sinews from his arms and legs to leave him helpless.

When Zeus defeats the Titans, because they cannot die he locks them in the immortal dungeon of Tartarus.

The Aloadae giants imprison Ares in a jar for ten months. He wastes away so that he is left without any strength but cannot die.

The god Zagreus is cut into pieces by the Titans and devoured. But Zeus rescues his heart and the god is reconstituted as Dionysus.

No. They may be injured, even severely but they cannot be killed. For example, in the Trojan war Athena impaled Ares, and he was extremely hurt but did not die. Also at one time Ares was locked away without food, drink, for more than a year by Giants but he did not die.

Yes

no, but they can feel the pain of death : Phersephoney for example

I don't think so. Maybe they can kill eachother...? I don't think they ever can die, though

No (as far as I've heard).

nope, they can be eaten but they'll just come out alive eventually