I beleive Dent survived. Yea, yeah the scrips says he broke his neck, but not everything from the script is brought over to the movie. The scene was plenty ambiguous. There was no confirmation of Dent's death from either Batman or Gordon. Gordon could have easily said "Harvey's prosecution died with HIM", but instead e said "Harvey's prosecution dies wit HIS REPUTATION".
They made a point in the movie to show how someone falling from that small height can survive. If they wanted to confirm he was dead, they should've used a higher place. Maroni landed on his feet and broke his ankles. Dent, we don't know how he fell.
Gordon could have faked his death, he faked his death in the very same film. A memorial service in the same movie for Loeb was held and you can see his casket. In Harvey's memorial, we only get a picture, no casket. His fate was purposefully left ambiguous so they could easily bring him back if needed. Nolan also said that he didn't like killing off villains and the original plan was to have Two Face as the villain in the third.
Also, it would be perfect sense for Two-Face to be both "died as a hero" and "lived long enough to become the villain". He is not one but both. They could have very well stuffed him in Arkham in hopes of rehabilitating him.