One would ask if those would still be recognized, and what obligation a Federation Starfleet officer would have to bring in such a person?
Arguably, the same obligation that Kirk felt when there was reasonable suspicion that Kodos the Executioner may have been identified in "The Conscience of the King." Whether they were murdered on Tarsus IV or Earth in the Eugenics Wars, the mass murder of human beings is the same crime.
Problem is, those organizations haven't existed for centuries. Second, you would have no witnesses to what happened. Third, Spock called records of the period, fragmentary.
It would be tough to prosecute someone for three hundred year old crimes.
The government of Israel didn't exist during the Holocaust but still captured, prosecuted and punished Adolf Eichmann for crimes against humanity. And, unlike Kodos, Khan openly admitted his identity and the complicity of his actions in front of witnesses ("We offered the world order!").
Q states during one of the courtroom scenes in "Encounter at Farpoint" that the United Nations had been abolished by 2079, and their laws and regulations were no longer relevant after that time. Admittedly this is Q talking, but Picard and Data both seem to accept the argument, and they're the two who are most in "court room mode" at the time. We might reasonably assume the same is true of the International Criminal Court.
Actually, Q's exact words were:
Q: Objection denied. This is a court of the year 2079, by which time more rapid progress had caused all United Earth nonsense to be abolished.
Since United Earth most definitely exists at least from the 2150s onwards, Q's argument doesn't exactly apply. Also, Picard and Data were arguing that "no Earth citizen could be made to answer for the crimes of his race or forbears." But that doesn't apply to Khan.
They wouldn't be asking him to assume responsibility for all augments, but to answer for his own choices.
There is no contradiction as Into Darkness is a different timeline than Space Seed.
The point of divergence in the Kelvin Universe is the destruction of the USS Kelvin. Everything in the timeline up to that point (i.e., the Eugenics Wars, the events of
Star Trek: Enterprise, the Earth-Romulan War, etc.) should be exactly the same.