What i really want to know is, if joker killed Robin then why hasn?t Batman killed him?
Batman by the time of BvS longer has his rule. How has he not already hunted joker down and killed him? I still think it?s because he himself is Robin.
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First you need to understand what state of mind Batman was in BVS, he was not killing like the Punisher as your comment seem to suggest, he just did not care if the goons die after pointing guns at him, its like if you surrender i will just punch you and brand you.
He did not kill the human trafficker for example( its Batman intro in BVS).
Now about Joker, you have to understand that Letto's Joker is not like the other adaptations that we saw in the movies, he is 90% comics Joker, he is Batman oposite, he likes mind games and he can fight Batman as equals, even in the mental level, this is my understanding about the character after Suicide Squad and JL.
Batman did not kill Joker after he killed Robin because Joker played by the rules to the extreme, he did not back down and fought "fair and square", he wanted to break Batman to a point where he couldn't go back, he wanted to be killed by batman, and thats why he did not die.
The last paragraph is my personal understanding about the character and about Snyder as a storyteller by his interviews.
If you want to understand more about the JokerXBatman, read things like A Death in the Family, The Killing Joke, Morrison's Batman, Frank Miller's Batman or the recent New 52\Rebirth from the comics.
In all of that there is someone questioning why Batman did not kill Joker and the answer is always the same, its something that Batman dont cross as absolute and something that Joker wants more than anything.
But i just remembered that in the Frank Miller TDK 2, **** Grayson became the New Joker so its possible Leto's joker might be ****Grayson, since ZS loves Frank Miller's Batman.