There have been plenty of stories where Clark or Kal have had to wrestle with their internal emotions--that's where great stories come from. He's been angry, lonely, depressed, without direction, and felt hopeless at various points. He's invulnerable but his friends and family are not--the point of a lot of Superman stories is that he's suffered emotionally or been tempted to cross some kind of similar line.
Absolutely! And also, children's stories can get very dark. But I also think there are limits to how far children's stories can go and still be children's stories, and Superman (I'm italicizing it to emphasize the difference between Superman the narrative and Superman the character) should broadly respect those limits, I think. Essentially I think the emotional baseline of Superman ought to be something like the emotional baseline of Paddington II, and the limits of its darkness should go about as far as, say, a Harry Potter book (Harry Potter being a really good example of stories that explore dark ideas but always bring it back to an emotionally safe place for the children in the audience). I don't think it would be appropriate for a Superman story to, say, go to the kinds of places that The Walking Dead or a Cormac McCarthy novel would go. There need to be limits on how far and how dark Superman the narrative, and Superman the character, can go, or else you're not really writing Superman or Superman anymore.