We see quite a bit of the interior of the nu-Enterprise hangar during that launch sequence. You can see rows upon rows of shuttlecraft... stacked one on top of the next... along the sides of that big bay.
You can find that clip in several places, but here's one I found with a quick web search:
http://www.reelzchannel.com/trailer-clips/40209/extra-clip
When the gaggle of the much larger Academy shuttles landed aboard the Enterprise, and settled on the two levels of docking shelves, the general height and depth of the facility seemed very TOS-like; these shuttles just were packed much, much tighter, and the doors and the bay were possibly a tad wider.
I'm a bit confused by how you can be missing this... I mean, these shuttles are, each , significantly larger than the TOS shuttlecraft we know. The entire TOS landing bay is only just about exactly 3 decks high, to begin with, is almost exactly as wide as it is tall, and only about 1-1/2 or 2x as long as it is wide (not counting the area inside the door rotational half-circle area)
So, here's a little exercise - figure out the size of the nuTrek shuttle. Shouldn't be too difficult. Note that they're stacked "nose-in" along the sides of the bay, and note the distance from "centerline" (where the launching shuttle is) to the "shuttle shelves" along each side. That will give you a MINIMUM width for the hull in that region.
Now, note the number of shuttles seen side-to-side. And note the speed at which the shuttle is moving past them. And note that we can't see them when looking in through the open "hatch" (it's not clamshell doors anymore, it seems) of the shuttle bay.
Compare that to the entire secondary hull of the TOS ship. The bay we see there is easily as long as the entire original secondary hull. You could only park two of the nuTrek shuttles inside the TOS landing bay... and those, side-by-side, facing fore-to-aft (or aft-to-fore).
There is NO WAY that this bay is remotely like the TOS bay... which is really not all that big.
The bridge set was a bit bigger than the TOS counterpart - but this time, it was housed on "Deck 2", the wide deck beneath the topmost dome, as evidenced by the external shots of the combined front viewscreen/windshield... So no problems with fitting the bridge aboard even if TOS dimensions were retained.
As for rim windows, those always were of a misleading size and placement even in TOS and TMP. So once again, the saucer could be 1.5 decks thick, much as in TOS and TMP.
Nope... of course, the "nu-Enterprise" saucer is dramatically different from the TOS one anyway... the bulk of its volume exists in the disk itself. It's a MUCH thicker disk, with much smaller "extensions" above and below the disk, relatively speaking. If the TOS ship primary hull is 11 decks, and two of them are in the disk (not considering the undercut of course)... in the new ship, if it were (say) 12 decks, four of those would be in the "disk." Of course, looking at the window levels (and taking into account space above and below)... I think we're looking at a five-deck-thick "disk" with the top, bottom and middle decks being windowless and the in-between decks having windows. So, since the "bumps" above and below are almost identical to the "disk," I think that makes the primary hull 15 decks thick, with significant structure between decks (and the nu-Enterprise saucer being thinner, overall, than the TOS one, proportionately). A TOS-proportioned saucer, at the same deck height, would be more like 20 decks thick.
On the question of how Nero's 2233 intervention could have changed the ship "so much", I think there is no real problem. The TOS ship was supposedly the product of 2245; the STXI vessel was launched in 2258. Twelve years of time difference translated to a significant stylistic and technological difference in shipbuilding on Earth's 20th century; why not on Earth's 23rd?
This, I have no problem with...
We all know that high-profile projects tend to get more "chefs" involved in their creation. I have this sneaking suspicion that the nu-Enterprise, in-universe, had a group of design engineers perpetually badgered by non-technical types... counsel members and the like... to "make this kewler" or so forth.
As for the stardate issue, I think we could pretend that it was just coincidence that SD 2233 happened to match the year 2233, and SD 2258 happened to match the year 2258...

I mean, in the TOS "system" there must have been at least one such chance match, too. And if stardates are cyclic (or have more than four digits, of which the first ones are usually dropped), then there'd be a number of such chance matches.
I would have less of an issue with disregarding the stardate "gaff" (since stardates are effectively nonsensical anyway) if it hadn't been made such a big deal of.
FYI, I saw the movie weeks ago... when the "Alamo Drafthouse" here in Austin advertised a big-screen showing of "The Wrath of Khan." I wanted to go see that flick on the big screen again, and to be honest, I was really sorta ticked that they pulled the "switcheroo" and it ended up being a sneak-preview of the new Trek film. So... I've seen it, but technically I've never paid to see the new film... I paid to see "TWOK."