The heroes seem to have kept their code against killing from the Silver Age, while the villains have become more and more sociopathic and bloodthirsty. Today, Batman folds his arms and "refuses to kill" no matter how many deaths his inaction allows. Are we, as readers, supposed to admire that? The fact is, if a police officer refused to kill a criminal who was in the process of a murderous rampage through a crowded city, he wouldn't be lauded for his high-minded principles, he'd be fired for incompetence.

Batman doesn't trust the authorities to keep crime under control in Gotham City, so he acts as a vigilante. But when he catches the criminals, he suddenly does trust the authorities to sentence them adequately and keep them imprisoned? It doesn't make sense. The "I will not kill under any circumstances no matter what" schtick that DC does with Batman isn't noble, it's noxious. If a police officer refused to shoot a mass murderer who was in the process of killing more citizens, he wouldn't be lauded as a hero, he'd be thrown off the force.

Also, who is stronger, Superman or Hulk?

Shoot to thrill.
Play to kill.
Too many women with too many pills.

Interesting, thanks.

Makes sense when his rogue's gallery are pranksters and misfits. When they cross the line into serial killers and spree murderers?

It makes sense in either scenario. He loses his police partnership whether he murders a prankster or a serial killer -- murder is murder. Cops aren't allowed to kill serial killers whenever they want, so I doubt they'd let Batman either.

Of course I'm not advocating that Batman go all Dirty Harry on the Joker or any other villain, per se. I'm more arguing that storytellers shouldn't resort to the cheap gimmickry that is "psycho killer Joker."
The moment the Joker becomes a psychotic killer in a serialized, ongoing story is the moment that you've officially "broken" the narrative. That is, if the Joker kills dozens of people every time he escapes from Arkham, and the Joker always escapes from Arkham, then Batman loses even when he wins. Because the ironclad laws of story demand that the Joker will return and that next time he'll do something even worse to raise the dramatic stakes. Batman will always triumph, but it's a hollow victory because the Joker will always be back. Therefore, Batman's moral code comes across as preening and self-righteous. Basically, Batman lets people die so that he can feel good about himself.
Batman stories simply shouldn't be written that way. It's a cycle of despair that inevitably leads to darker, almost nihilistic storylines and character arcs. The only way to avoid that is by NOT having the Joker be a mass murderer (or at least have Batman pretty much always succeed in stopping him). Batman: TAS pulled this off perfectly. The Joker was a credible threat, but he was never Charles Manson. And when he did cross the line in Batman Beyond: Return of the Joker it was the LAST Joker story in that continuity. It was a transformative event with real drama and real consequences for all the characters involved.
Psycho-killer Joker can be used very effectively in a "closed system" narrative like Nolan's Dark Knight movie or Frank Miller's Dark Knight Returns. But once you introduce that version of the character into an ongoing serial narrative... All you do is darken the stories and make Batman look less effective and less heroic with each new telling.
You can probably tell I have strong feelings about this...

But they are under certain situations, such as when the life of another is threatened. Batman doesn't even do that.