No, he dies in a flashback of the Batman Beyond movie, which while in canon with the batman animated series, isn't actually part of it.
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She falls off a cliff. But even then she doesn't actually die, because we see her towards the end of "Batman: Return of the Joker" in her 60s or 70s. And she's not into crime anymore at that point, so she probably dies of old age.
It's quite complex really: He was fighting batman at the top of a tower and batman was hanging of the edge. Then a joker helicopter came with a rope ladder. The joker got on the rope ladder as batman threw a bat thing with a rope at him. One end tied to the tower one to the Jokers leg. Then the helicopter flew up and Joker let go and so did the rope. Then he fell to his death.
Mark Hamill is still alive and doing voice acting and cameos. He's probably best known now as the voice of the Joker, which he performed for the final time for Batman: Arkham Asylum
Yes, he does die in batman begins because of the train crash.
There are a few theories, is was that The Joker once was an namless engineer that got fired from his job of a chemical plant to start a career as stand up comedian. He failed and got in finincial problems. Desprate to take care of his wife he agreed with to mobsters that offered him a part in a armed robbery on his old plant. He disquesed himself as "The Red Hood" to complete the illusion. He was than cornered by Batman, when he tried to tell him he wasn't bad, he fell into a vat of acid. The world belived him death, but he wasn't. The acid died his hair green and made his skin permanent white. Another is that The Joker once was a criminal named J. Napier, who fell in acid. Choose the one you want to belive.