Hmm... The timeline would allow for Carol to have joined
Starfleet here in a move that never took place in the Prime timeline - so the issue of her retiring in disgust wouldn't need to arise.
Apart from that, Daddy Marcus would probably just be one in a long line of hardliner Admirals in both universes; he just had greater means available in the nuMovies, thanks either to intel gathered from the Nero incident or to bloated budgets created in the aftermath of the Nero incident. If anything, the nuStarfleet might be "softer" than the Prime one, which is why Pike is so worried and eager to enlist hotheads like Kirk.
The same way Kirk knew: when the commanding officer finally figured out who Khan really was. The only difference in this case is the CO of that ship was a (more paranoid than usual) Section 31 affiliate and accordingly didn't give Khan unrestricted access to the ship's library computer.
Yup, that's how it would work after
they boarded the sleeper ship. But why would they board it? From a distance, it looks like an automated ore carrier, and there are no lifesigns (sez McCoy). There's the distress beacon, but why would that attract a S31 operative? Another ancient distress signal didn't attract Pike in "The Cage" when he had a much less pressing competing mission.