If Joker has to die at someone elses hand, then Gordon would be a fine choice. Or have it done like in the Harvey Dent situation at the end of TDK, where Batman ends up killing him to stop him doing something really evil.
I don't think that was intentional. He was just trying to grab the boy. Harvey resisted and then they fell down from that height.
As for the Joker, I don't think it would have been possible for anyone to kill him. In the Joker's mind, there are only two people in the whole world that have the right to kill him - Batman and the Joker. If someone other than Batman is about to kill him, chances are that the Joker will try to kill himself before that person gets the chance to kill him. This is at least true about the Joker in the comics. We don't really have solid evidence that Heath's Joker believes the same thing but I like to think that he would, especially after his "I think you and I are destined to do this forever" line.