Is Khan Noonien Singh running Starfleet in this timeline?

No, he’d have been deported to the Klingon Empire by now. The braying asshole whose shysters just extorted $16 million from Paramount is merely a student of his methods:

“I’ve done far worse than kill you. I’ve hurt you. And I wish to go on hurting you.”

To boldly go … where? To the bank for a cashier’s check, it seems. Hold the phasers, drop the shields, piss on the dilithium crystals and call in the Tribbles.

Beam me up, Scotty — there’s no intelligent life down here.