The Driver is a sociopath who thinks that he's in his own movie. That life is a movie. I know a guy like this. This guy has awarded himself the role of the rogue who doesn't conform to social mores. This is frustrating to deal with at times because of its disingenuousness, though any annoyance you give off is in a sense playing into this guy's role that he's created for himself, because of course you're upset, he's the rogue who doesn't conform to social mores.
It makes sense that Irene falls for The Driver. While not a sociopath, she is definitely introverted and quiet, like The Driver. No one else in the film takes as long to speak as those two (recall the scene when the police are speaking to Irene re: the death of Standard, when the cop says 'Can you...answer the question?').
All of the music in the film is being played in real time by The Driver. The love song fades when he shuts his door, the operatic song disappears with his car when he drives past Nino.
The Driver wears gloves every time he kills someone, except twice. The first is in the elevator, when the murder is witnessed by Irene. This scene signals their end, the real world clashing violently with the world he has in his head. The second is when he and Bernie stab each other, a scene shot in the shadows, the most 'realistic' killing of the film. You can see the gloves in his shadow's back pocket. He has re-entered the real world, his fantasy has not turned out the way he wished (Irene didn't fall in love with a murderer). The Bernie killing is preceded by the Nino killing, a scene in which he's wearing his mask from his stuntman job. This is the peak of his fantasy. He is the 'Real Hero' of the song. This of course juxtaposed with the fact that his mask is rubber. When the song plays again, at the end, he actually is both the hero (as he did, after all, save Irene and her child) and a real human being (the fantasy is over, he's not playing a character in a film anymore).
New layers every time.