"Father's Day" is a deep one. It's a loose adaptation of
New Gods #11, the final issue of the original series. Its biggest concession is one that's inherent to centering the conflict on Superman rather than Orion, and it's right there in the title: the original story is called "Darkseid and Sons".
Writers Mark Evanier and Steve Gerber (good choices; once again the showrunners hand a Kirby adaptation off to people who knew and worked with him and, in Evanier's case, were there when he wrote New Gods) do their best to preserve the father-and-son theme of the original story by adding Pa Kent to the story. It works well enough, and putting Pa's life on the line gives the story some personal stakes even though Superman has no idea who Kalibak is.
Nonetheless, it lacks the power of the original story, where two brothers fight to the death and ultimately their father steps in to save one of them, knowing it will mean the death of the other.
Of course the other difference is that the original story is an ending and the adaptation is a beginning. "Father's Day" is really Darkseid's introduction; we've seen him before, briefly, but this is really the first time we get a picture of who he is, what his deal is, what his relationship is to Kalibak and Desaad. Whereas New Gods #11 was the final issue of the series; it infamously doesn't provide a satisfying ending to the storyline, but it does have the sense, at least, that it's the end of an act, with Orion learning he's Darkseid's son, and both Kalibak and Desaad dying.
Darkseid
does vaporize Kalibak at the end of Father's Day, and it's pretty shocking for a moment there, but then he respawns somewhere else. Which is probably a mix of BS&P pressure and also, once again, this story being more of a beginning than an ending; it wouldn't make a hell of a lot of sense to kill Kalibak off in his first appearance.
As for Superman's encounter with Darkseid, that's a pretty straight-across adaptation of their first encounter at the end of
Forever People #1 (except, y'know, without the Forever People). And damn it's good.
Superman: TAS really does a fantastic job with Darkseid and the New Gods of Apokolips. (Maybe not quite as good a job with the New Genesis half of the cast.) Just really across the board -- the writing, the art, the casting. Michael Ironside is a perfect Darkseid. He's restrained, he's patient, he doesn't raise his voice. He makes for an excellent contrast to Luthor -- Luthor is defined by his insecurities, his need to prove he's better than Superman; Darkseid doesn't need to prove anything to anybody. He defeats Superman, literally with both hands behind his back, says "That is who I am," and leaves. (A riff on "Darkseid is," a catchphrase which I believe was introduced by Paul Levitz.) Michael Dorn is a fantastic choice for Kalibak, too, a borderline-feral warrior but with a sense of vulnerability and guilelessness; all he really wants is to impress his father, which makes him an easy target for Desaad's manipulation.
It's interesting to see the development of how the Fourth World has been adapted over the years -- Darkseid showed up in the later seasons of Super Friends but bore little resemblance to the version in the comics; Alan Burnett's spoken about the restrictions at the network and at Hanna-Barbera and how the DCAU shows were the kind of shows he'd wanted to make all along. (I doubt Kirby complained about the Super Friends version; he was just happy to get a creator credit and a royalty check.) Superman: TAS was finally a chance to get it right. And then Young Justice went completely buck-wild with the Fourth World stuff, introducing the DNA Project in the first episode, the Forever People not long after, and plumbing all sorts of weird corners of the lore like the Overlord (but ultimately having very little of Darkseid himself). And now we've got My Adventures with Superman bringing its own take on the likes of Intergang and the Newsboy Legion (and OMAC, which is Kirby but not Fourth World).
Quite the vindication not just for Kirby but for Jenette Kahn, the person who looked at the Fourth World books and saw their potential where her predecessors at DC had refused to. It's been immensely frustrating seeing the current owners of DC (and the ones before them, and the ones before them) systematically dismantle her creator-friendly legacy, as people like Bill Willingham will attest. But for a minute there, DC was managed by people who understood what they had and treated it -- and, more importantly, the people who made it -- with the respect they deserved. Or at least made an effort to. Obviously they didn't please everybody (see any interview with Alan Moore), but I've seen a lot more creators praise Kahn and Levitz's leadership than criticize it.