Thank you. I had to wait 10 months for your reply but it was worth it.I've now watched all eight seasons, straight through.
My final verdict:
House the character is INTJ.
The actor playing him is ENTP.
The writer(s) of the show are very INFJ.
He is not an extrovert. Most of his time is spent thinking, alone.
His intuition is Ni, not Ne. He doesn't brainstorm himself. His team brainstorms for him, and Ni picks the idea(s) that look most productive to him. When he diagnoses solo, he sees a single clue (Se), and synthesizes that clue into an overall explanation of everything (Ni). (A lot of his synthesis has an INFJ feel to it, positing personality traits of the patient, etc., due to the writers' INFJ bent.) He tends to jump to conclusions and insist that he be allowed to follow through on the diagnosis, which annoys his fellow (usually NTP-ish) doctors to no end.
The extroversion I do see is of the Te variety. Writing symptoms on a whiteboard and staring at them and crossing them out is a very Te activity. The brainstorming from his team is also dealt with in a Te way: an idea is quickly determined to be good or bad, and once the ideas are all out there, a decision is quickly made. Ti would tend to hem and haw a lot more. His team hems and haws, but House makes decisions quickly and decisively. The team is worried about getting the diagnosis correct, frets about possibilities; House stays quiet, then makes his diagnosis/prescription, and doesn't let the arguments last beyond that point. If he makes a mistake, that because a new datapoint and he reevaluates via Ni. Often, he deliberately makes a choice that he knows might be wrong, because HOW the choice ends up being wrong will provide additional data for the diagnosis. This is a Te approach in that making mistakes is deliberately used to figure things out in an iterative way; Ti prefers to avoid mistakes.
When I see him relate closely to individuals (the girl in the 3rd season with whom he spent a lot of time in a single episode, the woman he met in the asylum in season 5, Cuddy in season 6/7), the emotion is almost entirely on an Fi level.
And his stress modes exude Se, whether juggling, playing action-oriented video games, drinking, popping Vicodin, riding a motorcycle, whoring, or driving a car into a house.
None of this is 100% consistent, of course, due to the influence of the writers and the actor, but overall, what House does and how he does it says INTJ. The writers' INFJ-ness add to the overall INTJ feel by giving House very Ni psychological experiences, especially w/r to talking to imaginary people, and I suspect the inferior Se implied by the writers' being INFJ reveals itself in House's stress reactions. I also see an INFJ influence in the psychological nature of House's various pranks, and baiting of his team to pick on their current psychological insecurities.
In spite of the inconsistencies, what remains consistent is how his psychological breaks happen. They're very Ni-ish, with Se stress relief, and there is an overall ESFP vibe to how he becomes completely reckless.
I agree with you (intj for House - Te,Se,Fi and Ni) and you have written it quite convincingly.