A) The Joker was taken into police custody at the end of the movie. We can assume that. Given his behaviors throughout the film, we can assume he ended up in Arkham Asylum.

B) Yes, Harvey Dent was killed by Batman in the standoff. They held a funeral for him at the end of the movie. They wanted people to think he died a hero, so not all hope was lost in their eyes, given that Gotham’s other hope, Batman, was hated.

C) Yes, a man hunt is issued for Batman at the end of the movie. They blame the murders committed by Harvey on him so Harvey can be seen as a hero, and Batman who was already hated can take the bullet. As you could see at the end, they destroy the bat signal, much to Gordon’s discontent. If you’ll notice in the film they call Harvey the “White Knight” and Bruce the “Dark Knight” at the end.

D) Rachel did die. When the Joker took both of them hostage, he told Batman their locations, however told Batman Rachel’s location was Harvey’s and vice versa. He went to save Rachel, but instead saved Harvey. She did die in an explosion.

Any other questions? Sorry for rambling.