Why does Superman scream at the end of Man of Steel?

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.

Also....

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.

Superman screamed because General Zod was his last hope to learn who his people were. Superman already felt as if he was alone and did not fit in. It seemed like he was hoping Zod would stop and come to his senses and maybe they could one day get along.

I would say there are five reasons he screams....

1. He was forced to make a choice that his Father (Jonathan) placed before him let innocent people die or kill Zod.
2. This is the first time he's ever had to take a life and he feared it would change who he is and the Choices would make him.
3. Zod is the last connection he has to Krypton with the other ship sucked into a Black hole (Which mind you parts of that are like how?) Zod is the only remaining Kryptonian Clark knows. And that means Krypton is now truely gone.
4. It raises the questions of weather Clark is capable of making the same choice in a Human life or if he is above any oversight (Which we see again in both Batman v. Superman and Suicide Squad)
5. Jor-El intended Clark to be a bridge between the Kryptonians (if there are any others) and Humans. Would you be willing to Trust Clark even if he's not Zod

Superman screams not because of the terrible feeling of his first kill. His fight with Zod killed thousands of innocent bystanders and he doesn't pause to mourn.

Superman screams because he realised by killing Zod, he became the last Kryptonian in the universe and his hope for reviving Krypton is crushed to the ground.

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.

1. The killing.
2. The enthusiasm of it.
3. The guilt because you took a life.
4. The acceptance and resolution that what you did was necessary.

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.

My first impression was that he screamed because he had just killed the only last Kryptonian alive apart from him no matter how bad or good he was. And to the very last moment he probably thought he could've changed him or reasoned with him just like his father did.

How would you feel if you were the only Earthling raised on a different planet and you just killed the last of your kin ever exist in the Universe, especially that only very recently you found out where you come from and who you really are?

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.