I get the concept of an oil rig supposedly stirring up some leviathan from the deep, but I can't see how that can be connected to a satellite falling out of orbit. Even if said satellite didn't just burn up as most do, it would have been melted into an unrecognisable lump before smashing into the ocean. If it managed to survive even that, it would sink to the bottom where presumably it could be recovered quite easily by a deep sea salvage operation. No drilling would be required and it would take a lot longer to drag a drilling platform into place than a salvage ship anyway, so why bother?
In any case, the big explosion at the start clearly wasn't the rig as that was mid-Atlantic. It presumably wasn't the oil tanker either as that seemed pretty intact, so what was it? Also, given the timing of when the Statue of Liberty's head arrived on the scene to when we first saw the monster (a few seconRAB) it looked like the monster had snapped the head off, carried it onto Manhattan and then chucked it at the tall building which had clearly offended it in some way!
Don't get me wrong, I thoroughly enjoyed the film, and actually would rather none of this background stuff existed. But as it does, and as I enjoy picking holes, I will! Let's face it, if Mr Abrams spent so long putting all the viral stuff together, he should have made damn sure it all made sense!