Metahumans often, though not always, have their powers activated by a lethal event which determines the type of power they have. For example, the Flash gets super-speed after being struck by lightening during a lab experiment. Captain Atom got his powers from an atomic experiment. The term is basically an excuse to retroactively tie together all the disconnected "Freak Lab Accident" super-hero origins. How did they survive the accident and get powers? They were metahumans, the accident activated their metagene! Of course, some of them do just get Puberty Superpower, but not all. So no, Superman is certainly not a meta, he's an alien first and foremost, and Flash? He isn't.
As mentioned below with Luthor knowing more about Metas and how their powers operate than the other members of his trinity despite not having any powers, Superman likewise has been dealing with other Metas, despite not actually having powers any other Kryptonian on Earth would have, for far longer than Wonder Woman (since other Amazons are effectively 'normal' for her) or Batman. Anyone who actually knows more about Metas than Superman is probably a Researcher/Scientist like Luthor, and would be a Gadgeteer anyway, thus still defeating the premise of a Meta Mentor being a Meta. You really would need someone who's Meta power is understanding Meta powers which, AFAIK, does not exist in DC but would be very Meta.
Meta Villains: Lex Luthor? He doesn't have powers! He'd make much more sense as Gadgetry! Bane, or Poison Ivy perhaps?
Stretchy, but this version may be the kryptonite-steroid-pumping madman.
How about he mentors the metas because he knows more about their powers than anyone in the society?
This seems to be what he's going for. Most of his missions involve researching or manipulating metas, even if he isn't one himself.
Gadget Villains: The Joker? He's not got powers, sure, but he's not really gadget-oriented. Maybe a little...I'd personally replace him with Lex.
The mentors seem to be counterparts of each other. Supes, Lex. Batman, Joker. But I agree, the mentor titles should be a little different to suit the mentor or the mentor's should be changed.