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.
Yes, he did really die. This is most evident from the funeral held for him at the end of the movie, where James Gordon holds his obituary speech. As soon as he fell off the building Batman and Gordon examine him and shortly after the police arrive. There is certainly no way for any shady goings-on to take place and someone would notice things like a switched corpse or stuff like that, first and foremost Gordon himself.
And in fact Harvey Dent's death is a major factor in the ongoing storyline, since it is the basis for the "Dent Act" in place at the beginning of The Dark Knight Rises, the law that allowed the government to imprison criminals without a chance for parole. And it is also a major factor in the development of Bruce himself, as it is the culmination of the Joker's struggle to make him break his One Rule and a primary driver behind Batman's long exile afterwards. Without an actual death, these plot points and their meaning lose much of their weight.
The reason why you think he is actually supposed to become a regular villain to him is because you understand the Batman universe as a completely interconnected franchise, which is not the case. Christopher Nolan's Dark Knight trilogy (comprised of Batman Begins, The Dark Knight, and The Dark Knight Rises) marks a complete and self-contained story retelling the Batman mythos in its very own way. Any other Batman works bear no significance to it outside of inspiration, they don't exist from a story-wise viewpoint. Therefore, it is no problem to let an otherwise important character like Two-Face just die or to ultimately
make Batman give up his job entirely at the end of The Dark Knight Rises.
Harvey Dent officially died in The Dark Knight. Not only has Aaron Eckhart confirmed it, but it is literally written in the script that he is dead, as written in this article:
Aaron Eckhart says Harvey Dent is definitely dead: "He is dead as a doornail. He ain't comin' back, baby. No," Eckhart tells E!.
"I asked Chris (director Christopher Nolan) that question. He goes, 'You're dead.' Before I could even get the question out of my mouth, 'Hey Chris, am I...' 'You're dead.'"
The issue is also made rather clear in the script for the blockbuster sequel, a copy of which was recently released. It's a long read, as far as scripts go, at 167 pages, but page 163 seals the fate of Gotham's D.A.
"Dent lies at the bottom of the hole, his neck broken," the script reads, then succinctly punctuating the death in capital letters, "DEAD."