Superman had my sympathies during that episode and Batman behaved like a real jerk. He even tells Superman to "cry me a river." Oh, I see Batman. It's wrong to dismiss your own failures and mistakes but dismissing other people's failures and mistakes is fair game? Batman should consider himself lucky Superman didn't decide to demonstrate the effects of Kryptonian heat vision on the human brain.
As for Darkseid, he was definitely one the better characters in the DCAU and his relationship with Superman was one of the most engaging stories the DCAU had to offer.
The only upside I can see with Darkseid's relationship with Orion is that they are related by blood and that they are destined to kill each other. That's it. The only thing binding them is an ancient prophecy. In my opinion, Darkseid's relationship with Superman runs deeper and gets a lot more personal because of the things Darkseid does to Superman, such as brainwashing him or Supergirl. Superman: The Animated Series established a pretty high standard on how to portray both Superman and Darkseid.
As for Superman taking a place among the New Gods, it's not a bad idea on paper, but it would feel as if Superman would be rejecting his place on Earth simply because hanging around with people with the same powers and abilities would feel "easier." Yes, he can be with people who have the same responsibilities and burdens as him but the people of New Genesis are not Lois, Jimmy, Perry, Jonathan or Martha. They are not even Batman, Wonder Woman, Green Lantern or Flash. Superman doesn't have the same personal connection with them, the same way he is connected with the people of Earth.