In the final confrontation with General Zod in Man of Steel, Superman breaks Zod's neck just before Zod kills some humans in the corner of the train station with his heat vision. After breaking his neck, Superman screams in agony. Why is he agonizing? Is it because he was forced to kill Zod when all throughout the movie he was a pacifist? Or is it because he realizes that he has also killed the dream of his father to be a bridge between humans and Kryptonians?
I think you're astute in your approximation, you've surely already pretty much nailed it in your answer.
This was clearly a very emotional moment for the emergent Superman, not only for the reasons you've identified but also because Clark/Kal has just taken his first life.
He is presented as being your average, everyday American boy. To commit what is tantamount to murder, despite the nobility of his reasons, must be pretty horrendous.
That, together with the reasons you've already listed, is surely enough to prompt someone to scream out in anguish?
It's a very powerful way of humanizing him, allowing him to be seen processing the consequences of his actions.
It's an effective way of contrasting him against Batman, possibly in preparation for the much anticipated Batman Vs Superman Movie.
Batman's great weakness was his unwillingness to take another life, it was his absolute rule.
Although the new films won't be part of the Nolanverse, its a fairly widely accepted characteristic of Batman that he doesn't kill. Superman has already proven HE WILL.
I recommend you read the Book "On Killing" by LTC David Grossman. Killing is not a natural act (nor jumping out of a plane...lol) and most soldiers spend time through basic training overcoming the resistance to killing. Its not like stomping ants. Clark was never conditioned to kill, unlike Zod where he was genetically made to not have that natural resistance. To take info from the book when you kill there are several stages.
Remember the man of steel may be Kryptonian but his morals and actions are very very human. He screams because the guilt of taking a life was enough to cause massive grief. Compare that with Faora who said: "For every human you save we will kill a million more", she has no resistance to killing.
He didn't feel bad when the others were sucked into the Phantom Zone, who is to know if they died in there, but he didn't feel bad. So there is also seeing the body and the realization of what he did being in the form of grief.
But notice he accepted it in the end.
The way it was done was on point. Now my geekness just shot up 1000 fold.....LOL.
Superman killed because he was a new hero who was unfamiliar with his powers and was unable to come up with a better resolution in that instant. By no means should this be taken as an indicator that Superman is now a proven killer. HE'S NOT.