As to what happened with Carter, the last thing we see before switching to the Kate vs. Sander-Thing action is that Carter has heard something, presumably the Thing stalking him. When he makes a reappearance, Kate has already made the Thing swallow the grenade, so it would make more sense to act human, lure her into the snocat where she can't run away easily and assimilate her, doubling the Thing's chances of making it to the outside world.
As for not mentioning the American base, that was just so Kate wouldn't drive in that direction for the sequel. And yeah, the way it just cuts away from her is annoying. They could at least of had her snocat's lights driving away into the storm until they were swallowed up or something.
My brother and I returned from holiday and found to our surprise that this movie which had only been playing for a couple of weeks had already left the mainstream cinemas and were showing in the independents - we decided to watch it quickly before it stopped showing!
Conclusions: As a sci-fi horror movie it stands up well, and is worth seeing on that point. As a successor to The Thing (1982) however I was disappointed. Frankly I was expecting to be on the edge of my seat for the entire movie, but that didn't happen (it didn't help that one of the creepiest moments - when Kate is lured away by the Juliette-Thing - had already been spoiled by the trailer). One thing I felt was lacking in the '82 movie (especially as I'd read the Alan Dean Foster novelisation) was an all-out shitfight between humans and alien, but it felt more like they were ticking the boxes from the '82 movie (that's how the axe got there, that's where the two-face Thing came from, the guy who commited suicide was, umm...when exactly did he kill himself?) I thought the "tooth-test" idea was very clever, and was hoping for more cleverness - especially after reading Peter Watt's short story "The Things" I was looking for some insights into the creature. Or a twist like the female protagonist turning out to be the Thing, and in a way that stands up to scrutiny afterwards. And the Thing in it's 'natural' (or previous) form hiding under the hut was good - the well-lit CGI later versions less so.
There's nothing about it that's bad, just not as good as I was hoping.