Spoilers yada yada
To people wondering why Batman saved Dent and not the girl, he wasn't trying to save Dent. Joker mixed up the addresses on purpose, it's a joke
Also, Batman just tackled Dent, he didn't mean to kill him or anything he just ended up falling. Also notice the coin landing face up, lucky side. Dent can still be alive or he can be dead, it's totally up to interpretation as of now
Batman doesn't kill, period, it doesn't matter what anyone does he will not kill under any circumstances