As for "My name is Khan" - He probably assumed everyone knew who he was. He's Khan, he's walking, talking, ego. Ego demands that the mere mention of his name would still carry weight centuries after his disappearance.
I could buy this: it's probably along the lines of "My name is... Hitler!", so he expected a more "ooooh!" response from the two (which he did't really get, and which probably irked him to no end).