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.