Does the Two-Face die in the final scene of the Dark Knight? If he does, then how does he become one of the long lasting rival of Batman?
Yes, he dies at the end of TDK. Two-Face is definitely one of Batman's long-lasting rivals, but not in Nolan's story arc. The films, like different comic book incarnations, follow their own timeline and this includes the origins, chronological appearances, and even deaths of villains. In the comic, Two-Face has been around since the forties or fifties, yet he was also birthed in the Long Halloween, which came out in the eighties. Though Harvey Dent is cured in the comic book story Hush, he is also a villain in other stories that are supposed to take place long after that arc. So, you see, different series' follow different time lines. Nolan made an executive, creative decision to only give Two-Face fifteen minutes of fame in his version of the story. He defended this decision, saying that an entire film about Two-Face would be boring because Two-Face isn't a very interesting character and doesn't have much depth. Some viewers thought that he just forced Two-Face (who he allegedly meant for the series' third installment) into an unfinished TDK because of Heath Ledger's untimely death, but he has said countless times that he had made the decision to include both villains in TDK early on in the writing process.
*When you watch the Batman Begins series, you have to accept the timeline as if no other incarnation exists. IE there are no villains beside Ras Al Ghul, Scarecrow, Joker, Two-Face, Talia Al Ghul and Bane. There is no Poison Ivy, Ventriloquist, Mr. Freeze, Deadshot, etc.