Reply time:
This. A million times this. How many lives has Lex Luthor ruined? Supes wouldn't even have to kill him. Hell, one good punch and Luthor's rendered quadraplegic. Problem solved. But no, Superman hasn't done that.
Let me explain the difference between Superman/Lex and Batman/Joker.
Lex Luthor does not blatantly do evil things in public like setting off bombs or killing a prison full of guards. Lex Luthor is a smooth talker, he's out in the open and does his dirt in the shadows, hence he was able to become the damn President at one point. Lex does not just murder people left and right for a laugh. In fact, Lex has done many things for the BETTERMENT of the people of Metropolis and the World, though he's distrustful, there's no reason to take extreme measures against him. Now, Lex has been a staunch advocate that Superman is evil and is not good for the people. Lex believes he is the true savor of mankind, not this "God" that calls himself Superman.
Joker is a monster that doesn't think twice about exploding a bus of innocents or derailing a train for the laughs. He has murdered again, and again, and in mass quantities at times. He fills the city and its people with fear at the mere mention of his name and he rattles Batman like no other. Joker is evil with no rhyme or reason, he's evil to be evil.
Do you see the difference? Lex is an egomaniac who's bent on proving himself the better of Superman in the eyes of the World. Joker is a man that kills indiscriminately and laughs in the face of his murdered victims.
Moving on to "why don't they" for Superman/Batman.
The idea of Superman is entirely different from Batman: Superman is a beacon of hope that stands front and center in the light. He does not hide in the shadows and people have complete faith in him. Superman fill people with hope, he makes people want to be better and does his best to up-play the greatness of humanity and the good in us all. People of Metropolis, and the World, see Superman as a beacon of light for Humanity. Now, imagine this man murders a "rehabilitated" Lex Luthor, the people are instantly distrustful of Superman and it completely turns the people against him on top of making the World want him captured/dead. Superman would still save the people but they would fight him at every turn and potentially makes people lose a bit of themselves.
Batman fills the scared people of Gotham with courage. He shows that people can stand up and take back this "lost" city. But Batman deals in the shadows, he already skirts the law at every turn and seems to be above the law. If Batman murders a monster like Joker, nothing is lost in the people of Gotham except feeling a little safer. The only thing lost is, potentially, something in Bruce, whether it be his inability to stop dealing street justice to these Super Criminals by killing them or that it devalues his crusade or what have you.
TLDR: Imagine your hero, an infallible man in your eyes, murder/severely injure someone because they "make his life hard". Now, imagine this man that makes people feel safe, a cop or what have you, kill someone that has murdered hundreds because the murderer felt like it.
