The gods in some of these systems have souls themselves, qualitatively identical with our own. Often, as I have heard, they too are mortal and subject to rebirth. This is especially true in Jainism and Buddhism, and I have heard this claimed by some in Hinduism too. In the varieties of Hinduism that I have read about (mainly Vaishnava) the gods have their source in God, or the Supreme Being. We can see this in Bhagavad Gita 10.2, where Krishna says:
>Neither celestial gods nor the great sages know my origin. I am the source from which the gods and great seers come.
This source spoken about above is the ground of all being, it's eternal and necessary reality. These lesser gods may be extremely powerful or live for millions of years, but they are not omnipotent or anything. These are the ones that people have a chance of being reborn as in these belief systems. You can't really become the Absolute though. The true self / atman originates from God, and are one with God, but are yet different from God. We're of the same substance, but finite and contingent.
I'm not entirely satisfied with this response but after rewriting it a half a dozen times, this is at least passable in my mind, and hopefully it will clear up a few things. I pulled most of the info from Hindu stuff as it is likely clear, I didn't really even touch stuff in Europe. Also here's a video from a guy who basically says what I am saying here on the gods. It's worth watching:
Read the response above you in this post and it might clarify a things for you in a roundabout way. They definitely shouldn't be described as omnipotent, but were extremely powerful, far more than any (((Marvel))) character would be.
It's worth noting that in some philosophies though Zeus was described in such a way that he hardly resembled the Zeus of myths. For example, just look at how many Stoics spoke of "Zeus":
>For Epictetus, the terms ‘God’, ‘the gods’, and ‘Zeus’ are used interchangeably, and they appear frequently in the Discourses. In the Handbook, God is discussed as the ‘captain’ who calls us back on board ship, the subsequent voyage being a metaphor for our departure from life (see Handbook 7). God is also portrayed as ‘the Giver’ to whom we should return all those things we have enjoyed on loan when we lose close relatives or friends who die, and when we lose our possessions through misfortune
>Stoics hold that the mind of each person is quite literally a fragment (apospasma) of God (see Discourses 2.8.11), and that the rationality that we each possess is in fact a fragment of God’s rationality;
Marcus Aurelius can be found saying similar things in his Meditations, where the governing principle of the world-order is variously referred to as Zeus, Nature, the Whole, Law, Fate, etc. He says like Epictetus that our rationality is a fragment of God, but he also talks about perfect gods which are concerned with human welfare, are able to read one's thoughts, never die, etc.