I disagree that House always solves his cases with Ni, or that even if he did that would make him an ENTJ. I just don't see him as a dominant Te at all. The man drips of Ne dominance to me.
That's the reason I typed Dr. Cox and House differently: Cox wont directly admit it, but he really does enjoy being the alpha-male figure and managing his underlings and telling them what to do, whereas I don't see House as really caring all that much. To me Cox just has a much stronger NTJ strategic intelligence, whereas House doesn't really care and has the NTP problem-solving interests.
Other thoughts: I think Fe in the third place makes sense for House, given his fascination with other peoples' personal lives, although that doesn't disqualify ENTJ at all. Also, the way House controls his team and their morale is also extremely reminiscent of Keirsey's description of NTP informative communication style trying to manage people.
But I agree, ultimately he's impossible to type, but it is apparent that he has high levels of Ne, Ni, Te, and Ti. The question is just what order. (I'd argue Ne > Ni = Ti > Te >> Fe)
Others: Elliot seems to direct extreme amounts of energy inwards, hence the introversion. But I guess she could just be (comically) neurotic, too.
And yes -- Molly was the psychiatrist!! She was Heather Graham's character for several episodes. She had the amazing ability to completely read people, was extremely intelligent, but was still very silly/funny and would playfully outsmart others to make her points. I've found one of my favorite subplots of hers on youtube:
YouTube - Scrubs - My Common Enemy