The irradiated bills was a ploy of Batman and Gordon to track down the mobs hidden money. The bills were marked with an invisible radiation marker that batman could later scan to confirm it was his marked bills - It was just a matter of letting it loose on the public and see where it ends up - hence the scene when batman appears in the volt and scanning the money (It is obvious that Batman giving Gordon the marked bills happens before the film started)
About Dent - The Joker's goal wasnt really to kill Dent but get caught so that he could bust Lau out of the MCU. It may seem convoluted but its all part of the Joker's organised chaos.
And with Dent falling, you could argue that the way you fall can be all the difference (people can die slipping on a bar of soap) Batman does have a military grade padded suit that would have softened the blow but he does look hurt - Dent has nothing, plus the coin showing that it lands on the black side at least symbolically suggest that death has occurred.