Inbetween watching the film a second time I also went back and specifically rewatched 'Space Seed' and for anyone who remembers it's content there are some things which could explain some events in the film, and which could be construed as using the white space of the TV episode as the formation of the film.
Since the film is kind of like the alternate version of 'Space Seed' fused with TWOK anyway.
The first interesting point comes in respect of Marcus awakening Khan. In 'Space Seed', when the Botany Bay is found and boarded it's Khan's cryostasis chamber that begins to reactivate him automatically. As Marla McGivers noted, sometimes the leader of a ship was set to revive first to assess their situation. It would be perfectly reasonable to assume something similar may have happened when Khan was found by Marcus.
Either that, or Marcus somehow knew enough to know who he was.
Once Khan is aboard the ship Bones scans him and notes that 'something within him refuses to let him die'. That's never quite specifically explained but something in his blood perhaps? This could be writers exploiting vagueness in TOS. Something perhaps unique to Khan over the generic supermen with him. We'll likely never know for sure.
On the face of it, using him to design weapons by exploiting his natural talents seems odd, but then in 'Space Seed' Khan himself says 'I used to be an engineer' and expresses immediate interest in reading up on the ship's systems and technology. He also seems to figure out enough of how the ship works to allow him to take control of it and threaten the crew. So, he's in some ways a demonstrably quick study. Later 2D thinking notwithstanding.
Now, for sure that's not in the film (does it absolutely need to be - I guess people will differ on that) but since the writers will be assuming fans will be familiar with 'Space Seed' there would be no need to say that over again I think.
Transwarp beaming is probably being over-highlighted. It's still essentially a secret from Starfleet in general as they whipped it from Scotty and promptly Marcus gave it to Section 31 to play with. If it wasn't for Khan needing to steal it to escape Earth it isn't likely that Section 31 would make it widely known they have such a toy. In and of itself it's a bit of a convenient piece of tech to move people around but so were dimensional transporters between Universes in DS9 and transwarp conduits in VOY.
'If the Apocalypse starts, beep me!' - Buffy Summers
'The sky's the limit.....' Jean-Luc Picard, 'All Good Things'
courtesy of Saquist