Khan has to be dumb because the script calls for him to be dumb.
All throughout TWOK there are indications that Khan is not exactly of sound judgement. 15 years of Hell would do that. It's less him being stupid, and more him being driven by a rage and a compulsion for revenge that is clouding his judgement.
*Where exactly did he think the rest of the Regula One people went? Where did he think Genesis went? Khan didn't ask Terrell or Chekov how the rest of the Regula One crew might have escaped? He knows there's a Carol Marcus yet there isn't one when he arrived.
He may not have known Regula One even had a
transporter, and he was pressed for time. He tortured and murdered everyone in a fury when his questions were not answered.
*Khan says in Space Seed that he would need help to run a starship. Yes he's smart but he only has like twelve people. Plus the technology is completely different than what he experienced when on the Enterprise.
And yet in Space Seed he knew the ship well enough to change the settings of key systems. His need of people was likely due to the ships size more than anything. He was more than willing to kill the entire command staff "Each of you will go in there. Die while the others watch!" More likely he just wanted the fastest route to total control, and that could be facilitated by the command staff.
Arguing the technology difference is moot, since a lot of that depends on how you take the visual changes between the TV show sets and the big screen film sets. The technology was very likely not that different in terms of how it was
supposed to function.
*And Khan didn't need to go down to get Kirk, he could have simply beamed him up.
I never said he did. I was referring to Khan not going down to the transporter room.
But again, as far as killing Kirk is concerned, he had no intention of doing either because it isn't what he wanted to do in the first place. He wanted Kirk to suffer just as he had suffered. That was his whole motivation. Leave Kirk to rot, destroy the Enterprise, and leave with Genesis. It's incredibly straightforward. You might find flaws in it, sure, but Khan not killing Kirk outright on the moon is not one of them. It's fully addressed in the movie as to why Khan does what he does there.