Norway Productions was just the name of Roddenberry's production company. Star Trek used three or four effects houses during the run of the series, so it's just a question of which one did the new effects shots for "The Doomsday Machine". And since the effects houses were across town from Desilu/Paramount, it's entirely possible that Jefferies never even saw the model until it appeared on screen, but that might be pushing it. It's likely that someone else built it, and considering the nature of the project, it could just as easily have been someone's kid and not someone actually on staff.
As for the registry number, the sole concern was so that it could be easily discerned at a distance, i.e., on a small tv screen (certainly small and horribly low resolution by our standards), so yeah, 1710 still looked too much like 1701.
And why the terribly inaccurate AMT model? Compared to mucking up the eleven footer (a very expensive piece of hardware) for the sake of one episode, sending a flunky out with a couple of bucks to get a store bought model and slap it together is not only an incredible bargain by comparison, it's also a major time saver (which is also a big money saver). And for the purposes of the show, it was close enough for government work. If someone got snarky about the differences in the details from the Constellation and the Enterprise, they'd probably come to the same rationalizations we have, using the aircraft carrier analogy, and say you're perfectly welcome to imagine that the Constellation is either a variant of the Constitution class, or it's an older class, whichever works for you.
I prefer the older starship class option. It gives the fleet a bit more of a sense of history, it fills the gap between the Daedalus and Constitution classes, and goes a looooong way towards explaining that freakishly low registry number.