I forgot about the spaceship!
But I still think the Thing copies aren't aware they're copies. The Thing might be able to take over, but I imagine the human brain part blacks out - after all, why not? It's the perfect cover.
Blair trying to get back in really depends on when he was infected, as it's one of the film's great ambiguities - was he the Thing then? If he was, did he know? Since he was happy to dig a tunnel and build his little ship, it doesn't seem to matter to it whether it gets let in or not.
The Thing is such a perfect imitation it even copies a shitty heart that compromises itself, which is why I think the facsimiles are unaware of their own nature. The Thing consciousness doesn't operate in our brains.