Originally Posted by The Joker
I think when he says "Do you want to know how I got these scars?" he's saying "Do you want to know how I became the Joker?". Why else would he adopt a clown look and say "Now I see the funny side. Now I'm always smiling". His scars are the big perma smile he's always wearing as a product of something terrible that happened to him.
You're dead right, the multiple stories was a nod to The Killing Joke. Heath Ledger himself said TKJ was handed to him to read as part of his prep for the role;
Most of Joker's scheme in TDK is a love letter to Joker's twisted experiment on Jim Gordon in TKJ.
Batman essentially says the same thing to Joker when the ferries don't blow up; "What were you trying to prove, that deep down everyone's as ugly as you? You're alone". You could consider Harvey Dent the Jim Gordon of the story in terms of Joker trying to prove someone really good can be broken, only in TDK's case Joker succeeded, whereas in TKJ he failed to break Gordon.
Exactly right in all areas. But that's to be expected coming from you in regards to this topic.