Re: The Joker's background?
My take on this is that none of the two stories the Joker told were true, on any level. Some subtle moments in a few scenes (when Batman calls him on his true motives or when a mobster would call him 'freak') reveal a man that may be perfectly sane and remembers his true origin, yet also hides a certain vulnerability. He also kept licking his wounds all the time, like a reminder, and he was the first to bring them into conversation ("Wanna know how I got these scars?"). Basically, he was obsessed about his own story, but wouldn't share it with anyone.
He seemed eager to tell traumatizing stories to some people, though. He had one for Rachel about a wife who abandoned him, and had one for Gambol about a violent father. Considering the listeners, those were very interesting choices. My guess is that he just had a story prepared for diferrent types of people and Gambol, like that elderly party guest, was a man of auhority that was standig up to him. When Joker says "... ad I hated my father" I sened he was going to start telling him the fake father story, maybe one that resonated more with the guy, designed to traumatize him. But then Rachel stepped up, so he changed his routine. Maybe the stories were made to make the one listening feel guilty about them. It's a great coincidence that Rachel's story was about a woman that said something to a man, then the man does something crazy to be with her and she leaves him for that. It has an astounding resonance with Rachel's subsequent development! Maybe the Joker got to her and inspired her a bit. Maybe the Joker was just a terrific judge of character.
But I don't think he's the kind of guy who would reveal the tiniest bit of info from his past. It seems that his true origins were too precious to him.