The root conflict of the story isn't just about the proper way to deal with the worlds problems, it's about the influence that people with power have on society.
Lois Lane on bad people deserving to die might just be seen as the writers trying to make her be a little at odds with Clark's view, until you realize her father is in the military and that she would have grown up with such a viewpoint, then it makes perfect sense.
Just to point out, Lois says that 'sometimes' she feels that way. She seems to mean it as a 'heat of the moment' or 'in a moment of weakness or anger' thing. She's actively trying to stop the crowd from supporting the Elite when they hold Atomic Skull's impromptu execution.
Manchester Black proves himself capable of fighting on (mostly) even terms with Superman during his Beware the Superman phase. If Superman hadn't simply been acting, then no matter who won the battle, it's safe to say that the entire world would be several kinds of screwed.
Not really, Supes was still clearly holding back. Had he genuinely wanted to kill the Elite, he would have beaten them, taxidermied them and put on an insulting marionette show with their corpses in the span of 2 seconds.
You can tell that Supes didn't ever really use his super speed other than to dodge attacks. If he really wanted Black dead, he woulda been punching alot faster.
Yeah, but it's obvious that they were doing damage to him. And Manchester was correct at the end that Superman could barely stand after his last telekinetic blast. You gotta remember, Superman is incredibly strong and can take a lot of damage but he's not invincible.
Keep in mind, Superman didn't beat any of them through brute force. While the comic version showed it more, all of the Elite except for Menagerie had the power to beat Superman on their own. The comic version of Coldcast was explicitly said to have far more power to throw around than Superman (although, then again, hyperbole is thrown around quite a lot in the comics). Even if he was holding back, during the head-to-head part of the fight Superman was still injured far worse by them than the Atomic Skull. If they'd taken the fight seriously instead of treating it like a Payperview event, things would have gone very differently.
But then, if they were the type to think things through and treat their enemies with respect, it would have been a very different story.
Perhaps not so... Superman can break the planet in half and has a top speed of 137.8 times the speed of light (without sun dip; with one, his potential could very well be infinite). Also, Superman may not have had direct sunlight contact when he went up against Black (and he probably didn't have time to pick up extra rays beforehand). Finally, after a certain point, the Elite are clearly terrified and ready to go all-out against Supes. Superman could have dropped Black without saying a word to him, burning out his eyes with a single blast of heat vision.
There's a difference between Superman the members of the Elite: Supes can take as much as he can dish and move so fast that even hitting him is an awesome feat by itself, the Elite can't. And that's what really makes Superman going evil so scary: he's nearly impossible to kill. The Elite, on the other hand, would be vulnerable to anything that can kill a normal man (in fact, had they won I can easily see a number of superheroes make short work of them, including Green Arrow using the boxing glove arrows just to humiliate them).
The methods that Superman used to fake the death of The Elite were pretty dangerous out right: he shot Pam with a dart that made her heart (appear to) stop, collapsed The Hat's lungs with a tornado, and carried Coldcast into space. While it's true he didn't actually kill them, those methods could've killed them anyway simply due to the forces involved. Superman could have accidently killed The Elite!
The Elite killed two nation's leaders - what will that solve? Are they going to rule those countries themselves? What if the people don't want to make nice with each other? Are The Elite going to set up a police state run by themselves? Or just kill both populations?
They Didn't Think It Through. It would have been quite amusing to show what would have happened... Especially in the comic, where Lex Luthor was the president of the United States and thus in charge of the country that usually leads the humanitarian interventions.
Superman reveals that he didn't actually destroy the part of the brain that gives Black his powers, but since the Elite is going to be de-powered anyway, what difference does it make?
That's part of Superman's point: Maybe the authorities decide to de-power them. But it's not Superman's place to make that decision.
Also, Superman's heat vision would have set fire to Manchester's eyes rather then just one part of his brain if he tried to do that.
Better question: if the ability to depower someone exists, and it is in the hands of the authorities, why are there repeat offenders of super powered criminals?
Perhaps, like with the Atomic Skull as a power source, some government officials believe they can use the Supercrimnals as a resource.