That the black hole deep inside Vulcan's core somehow made things complicated also doesn't make sense. It's a constant gravity source, there are no fluctuations.
But we already have to start from the assumption that this was no ordinary black hole. It was demonstrated early on that it was an interdimensional black hole that could reverse (or at least contain or lessen) the effects of a supernova, and established later on that it could pull in a starship capable of superluminal travel and multi-thousand-gee accelerations while having nothing but the mass of another starship to work with (but could be stopped from doing so by blowing up a few warp cores in the vacuum between the hole and the trapped ship).
"Anomalous" would be putting it mildly, then. We know transporters have trouble coping with things like thunderstorms and high winds and people who wave their hands about a lot when under transport. An unsteady gravitic pull would certainly work havoc with that kind of equipment.
They have only one Chief Engineer and then the position is up for grabs.
Is it? Scotty only started contributing when the ship was being sucked into the black hole. He was asked for ideas by the CO. He then asked the CO for permission to do a crazy stunt. He got that permission and pulled that stunt, which saved the ship. He didn't really command the Engineering Department - he just performed a single action for which he was eminently qualified (i.e. crazy enough to think of it).
If Captain and XO are gone, a cadet can take command and then keep it.
Well, it's what the Captain ordered, before his departure. I guess he could have ordered his fancy baseball cap to take over if he so wanted; he just chose Kirk, thinking he had the required expertise and the minimum competence required. Kirk in TOS often placed various Ensigns and Lieutenants in command, bypassing his bountiful supply of LtCmdrs and Cmdrs; this should have been no different.
while Kirk was indeed a cadet and shouldn't even been on the Enterprise to begin with
Starfleet may in fact have had such a problem with giving various independent starshipboard tasks to undergraduate cadets that it gave all of them a posthaste commission when transferring them to those starships. Thus, Kirk during the Vulcanian expedition would be a Lt(jg), just like the transporter control computer screen indicates (anybody taking the big package of command courses including Kobayashi Maru and graduating from the command track would be that, rather than Ensign), and not a cadet. Might be a temporary commission, might be permanent.
As for Kirk's quick promotion past three ranks between the end of the action and the final Academy scene, we don't really know how quick it was. Pike also jumped from Captain to Vice Admiral there, after all. Sure, there are indications that this may have been soon after the action bits: Kirk still somewhat bruised, Pike still in wheelchair, the same cadets in the same cadet uniforms in the audience, Spock Prime only contacting nuSpock after this ceremony. But we can easily argue that Kirk would keep on getting bruised for the next year or so after the action part of the movie; that Pike would have to spend considerable time in the wheelchair; that only Kirk would fast-track through the Academy and all the extras on the background would study for at least a year more; and that Spock Prime had better things to do, or wouldn't dare interfere before it was certain that nuKirk was on the proper path down the streams of time and that pairing him with nuSpock would work out all right.
I guess the biggest obstacle with thinking that the promotion came years after the action lies in Kirk still wearing a cadet uniform... But then again, we might be misunderstanding the uniform system.
Timo Saloniemi