My impressions upon watching TOS was that Kirk, Sulu and Uhura were about the same age, McCoy and Scotty were older, Chekov younger. Spock, being an alien, was a big question mark. "The Counter-Clock Incident" suggests he's the oldest of the crew, a teen when Robert April commanded the Enterprise.
Other than Spock's age (three years older than Kirk according to the deleted birth scene), the new film seems about the same.
I consider this Pavel Chekov to be a different individual, genetically, than the original. I assume the others still look the same in-universe despite what we see, just as Saavik or Tora Ziyal or any other recast character is supposed to; after all, Spock Prime recognized Kirk and Scotty on sight. But Abramsverse Chekov is not only four years older, he's also a Wesley Crusher-esque prodigy, whereas Chekov Prime never really seemed all that exceptional in the brains department. So I think of this Chekov as sort of a cross-temporal older brother who was given the same first name by his parents. Perhaps Andrei Chekov and his wife always intended to name their first son Pavel, but in this timeline they had a different son four years earlier.
That's a horrible explanation, IMO. Chekov is clearly supposed to be the same guy from The Original Series. It's much easier to say Chekov Prime underwent some relativistic time dilation that his alt-self didn't -preserving their different ages and giving a common birthdate.
As for his genius levels, he was considered sharp for a 17 year old in STXI - but maybe he was just an early starter. What's amazing at 17 is probably a little more mundane at 22.