I wonder if Kirk in the Kelvin Universe was able to keep command because Future Spock and Pike both put in a good word for him. Also Is it possible that the Enterprise isn't the flagship in the Kelvin Universe? Maybe the brass felt more comfortable giving him the job because the ship wasn't going to be used for the more important missions. I mean when the second movie comes out I was under the impression they were still evaluating him which is one of the reasons they were going to put Pike back in command.Nog did not go from Cadet to Chief Engineer in one episode. Nog stopped being a Cadet in Favor the Bold when he received a field commission and became an officer with the rank of Ensign. He remained an ensign for a year and a half and was promoted to Lieutenant, though there is no indication he was chief engineer. Regardless, going from Cadet to Lieutenant in the span of a year and a half during wartime is yes much more realistic than someone instantly being promoted from Cadet to Captain and given command of the flagship.
Harry is kind of pushing it, but Voyager is a smaller ship so it's not that bad. Dr. Bashir was a prodigy, having graduated second in his class in medical school, and as noted above DS9 was at first a backwater assignment.
It was an emergency situation. Had things gone normally, she'd only have been keeping the seat warm until a qualified pilot showed up, probably one of the background bridge guys. Unfortunately, the saucer just happened to crash while she was at the helm. And while everyone makes jokes about Deanna always crashing ships, the one thing they overlook is that in Generations, despite the fact she was sitting at the helm, Data had in fact taken over fight operations, during the crash it is he who leveled their descent so they would crash bottom-down.
As for Troi taking command I would still think command would go towards O'Brien or Ro or even the other dude their before giving it to a untrained counselor. I know O'Brien isn't a officer but they still felt comfortable with making him Chief of Operations on DS9 which means he would be giving commands to engineers who did go to the acaedmy.
Jason