Concerning Kirk's seemingly rapid promotion, I look at it this way,

During the US Civil War there were several instances where a junior officer was promoted to Colonel and command of his own regiment. Obviously a lot of this had to do with the high casualties and general desperate nature of the war that both the Union and especially the Confederacy faced.

In the Star Trek film you get a sense that a very significant chunk of Starfleet was destroyed by Nero and after the destruction of Vulcan and the impending threat to Earth, the situation definitely became a desperate one.

So Kirk getting promoted to Captain of his own starship after the exemplary courage and skill he demonstrated in stopping Nero, I'd say that his promotion was not far fetched.

I thought of all that, but I still think it was fast. I also think that Spock played just about as big of a role in resolving the issue, and he had more seniority, so would have been, dare I say it, a more logical choice to promote.

Plus Spock wasn't under investigation right before this all happened, and wasn't grounded and didn't get on board the ship through a loophole/trick.

Arthonon, the reason you spotted the frays and jumps goes with having written stuff. I'm betting the original script would have ran three hours and made a helluva lot better flow and sense, and the producer and director sat down with the writer and said "keep the first section of screen play to page 113, and then reduce the other 400 to 200. But the sword fight stays; also, we like the snow monster, but can we have a bigger monster eat that one, and then have it chase Kirk? Oh, and we paid Nemoy for the whole movie - let's see him in it."

On the Enterprise sets: you're actually not far off, in that they recycled most of the sets for different locations, and bunches of stuff was bought off the shelf or from online sales. It really worked well for DS9, but was too easy to spot on Enterprise due to a lack of clutter and the color pallette they worked with. I like the phasers, though, as they looked very prototype and flimsy.

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On Kirk's selection over Spock: then, as now, it is often not what one knows so much as who one knows....