Nothing in TWOK would have affected Chekov's career. Khan's escape was unfortunate but could have happened to anyone who visited Ceti Alpha V.
to Chekov, who should have known better. He was a starship navigator and bridge officer who had been aboard Kirk's ship when the Captain stranded Khan there! An obvious failure in Chekov's part, perhaps punishable.
Isn't there a reference to 'Admiral' Chekov somewhere in TNG or DS9?
Nope, that's in ST4:TVH.