...Chekhov, he could have been in another department.
Not to belabor a point that I think we're largely in agreement on, but Chekov was at least ten years younger than Kirk. As such it's almost impossible that he would have been serving with "Young Kirk" in the new movie if they were using ToS continuity. However, as you note:
Eric Bana's character is from the timeframe of Patrick Stewart's Star Trek: The Next Generation. It's time travel. JJ Abrams said this new timeline is an alternate one. The original Shatner-verse still exists. Meh.
Yeah, if they were going to do a reboot, they just should have done a reboot. Either that, or keep the original continuity as much as possible. This half-measure thing is unlikely to satisfy a lot of people.