So I just got back from seeing Cloverfield
and likewise the teaser trailer for the ST film. I can confirm that the dual blades in the nacelle caps do spin in opposite directions in the same way that Gabe's do.
But regarding the teaser in general, I don't think that there's any implied in-universe connection being made here. In fact, all the visuals seem to be pretty much a sight gag revolving around the tag line "Under Construction".
Everything in the trailer other than the reveal of the ship itself is modern and un-futuristic, the sight gag being that the full scale Enterprise is actually being welded together for the purpose of being filmed. It does look photo realistic, it is really cool, but I don't think anything in it should be taken any more seriously than the First Contact
trailer showing a giant Borg ship in the shape the Starfleet logo. It looks to be just one big titillating pun.