Does being Q zapped into an adult mean you can skip Starfleet Academy?
No. The Academy is basically school for officers. Skipping the Academy means enlisting and becoming an ordinary crewman (like Chief Petty Officer Miles O’Brien did). It was unclear how old the Benzite and the Vulcan were in “Coming of Age”, the episode where Wesley tried to apply for the Academy but failed. Also, I’m not sure how old Nog was when he went to the Academy–he just said he was an adult at that age by Ferengi tradition. So I don’t think age has much to do with it.
Nog was probably about Jake’s age. I know Aron Eisenberg was at least 25, but since Nog and Jake were basically in the same grade level in school in season 1 it’s safe to say they’re the same age.