They're talking about Batman's parents in the context of their origin stories - who made who. I think Joker realizes that killing someone's parents would only be meaningful or traumatic enough to make them "Batman" if it happened when they were a kid - i.e., a long time ago.
Adults, upon hearing that their parents are dead, don't dress up as bats and go fight crime. Therefore, Batman's origin must clearly originate from his childhood. Napier's intelligent enough to realize that.
See above. When Batman says "You killed my parents...you made me first," it's pretty reasonable for Joker to assume that Batman must have been a kid when it happened. Losing one's parents isn't a deep trauma for an adult.
I think if you go by just the screenplay, your argument would have more merit. But Nicholson's delivery of those lines make it clear he has no idea what's going on.