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If he wants to.
Many times (In the movies and TV shows) Batman has threatened to kill The Joker. He has never carried out his threats because it would violate his conscience. There are many comics where Joker does die and I believe there is an alternative comic called 'The nail' where Batman kills The Joker.
Yes because joker killed Jason Todd the second robin
Yes, the joker did kill people in the movie Dark Knight.
Because he wants to protect Gotham. The Joker relationship with batman presents an opportunity to examine the internal struggle between being a lawless vigiliante and being a defender of the law. The Joker on the other hand is a violent psychopath who uses the clown personnae to cause terror, death and mayhem. When Batman is seen beating up Joker, you begin to ask yourself "how far will he go?" Will he go too far and kill the Joker? If he does kill the Joker what does that make Batman, a criminal murderer himself? But he rarely (generally only in the really early comics), shown as killing criminals. Perhaps roughing up the joker is the writers way of showing that the Joker can take him to the "edge" but he retains his ability of self control.
The Joker.
The joker
Joker
Not in the comics universe.
WHAT?! Nobody can kill Batman!
No, but in the comics he sometimes loses it and tries to kill The Joker.