I would agree about "most" of the history and trivia, but not "all". Given his behavior in Chapter 40, it at least seems likely that he did not know as much as Harry about the Hallows at that time. This is understandable, as Harry has a Hallow and therefore cares more than the Defense Professor, who doesn't have one and doesn't have a particular reason to search for any of them. He wouldn't decline a chance to try the Stone, but he doesn't have much reason to believe it works as advertised and therefore didn't plan to seek it out. Now that he remembers "a peculiar ring he] saw on the finger of a man [he] met only once" ([Chapter 26), he is much more interested and probably knows more publicly-speculated information about the Hallows than Harry even if he doesn't know some of the specifics Harry learned in TSPE.

Similarly, the Defense Professor doesn't seem to care about whether other beings are sentient, so he probably does not know as much about the fairy tale of the tale of the Lady of the Flying Squirrels (Chapter 49), even now that Harry has mentioned it.