You sound rather cocky and almost personally offended by other people's opinions. Also, the name-dropping is pretty trite and unimpressive.
Littlefinger might be a more clear INTJ in the books, but he gives off an ENFJ vibe in the series.
Not sure what to say about Arya. INTJ is possible, but I wouldn't rule out ISxP.
ISTJs, like any other type, can be quite brilliant. No need to assume that's why people see him as an NT. I see him that way because of his interactions with people and his tendency to read between the lines and use it to his advantage. His foresight is superb and he seems to be always a step ahead. He uses the law and set standard for his advantage, but he doesn't seem attached to it. ISTJ doesn't make sense to me because then Ni would be inferior for him and I certainly don't see that.
I agree that Tyrion could be an introvert, but the reason I see Tyrion as an extrovert is not because he has sex and drinks--most of the characters do that. But because he's been able to endure more than a lot of the characters and has still managed to maintain quite a bit of vitality. And he frequently has to pick an unseen possibility inspired by his environment and go with it before having ample time to think it through. And he's okay with that. He's not okay with injustice, but he seems to gather inspiration from the outside world even if his relationship with it is love/hate. I believe his Fe function is higher than inferior, and I think his Ne is above his Ti because Ti users tend to be veeeeery thorough with an idea before putting it out there. In fact, I'm not sure I can rule out INFP for him because I've noticed instances that looked like Fi/Ne/Te.
Jamie is much more practical than Tyrion. He is the ultimate realist. He does what he has to do and uses whatever or who(m)ever is there to accomplish it. He's more of a doer than a talker, but he always has the right words to say when the time comes. I think you're the prejudiced one here who thinks sensors can't be brilliant and symbolic. Some great artists of multiple facets were/are sensors who utilize color, shape, and lighting to create nuance in order to convey something. Many of these people were still sensors. In fact, their preference for things to be tangible can be what motivates them to turn something abstract like an emotion into a painting or sculpture or film, etc...Jaimie is a deep, wise in some ways guy. Because he's lived his life without rose-colored glasses. He sees the reality in front of him and he analytical enough to be able to understand it profoundly. Sure, he was trained to fight. But when his arm was cut off it was symbolic of his life. What meant the most to him and what he felt most comfortable with. It was almost as if his dominant function was taken away, then replaced by his auxiliary. Now he uses more Ti.
Catelyn seems ESTJ to me. She uses Fi, not Fe, and is pretty 'active' and blunt. She always wants the truth and there's little evidence she's introverted.
Cersei is not practical enough to be an ESTP unless she is extremely unbalanced. She is more idealistic than she admits and is obsessed with control. She does not go with the flow. She temporarily pretends to, then complains and/or tries to manipulate the situation. I see a lot of Ni in her with unbalanced Te and Fi. She's almost constantly paranoid and sees how little things could cause a major domino effect, usually at her expense. She drinks and engages in sensory pleasures when stressed just like someone with inferior Se will do. She's ashamed of her emotions and doesn't really connect with anyone besides Jaime, because he is her twin. She's a worrier to the extreme and people who are Se dominant generally aren't constantly worrying about the future. When she gets a theory she sticks to it even if logic disagrees. She is supposed to be like her father, except less logical and more emotional. Well, if you go by functions, Tywin being an ENTJ and Cersei being an INTJ fits. Obviously, she is unbalanced and has paranoid and narcissistic traits, which INTJs are susceptible to. I don't see much Fe in her, because although she is charming and plays her role well, she's clearly miserable and is fine with the world being in chaos as long as she and the few closest to her are secure and in harmony.
Joffrey is hard to type. I don't see a clear preference for introversion over extroversion. I think if I had to guess though, I'd pick either Se or Fi as his dominant function. I'm not sure though because I think he uses immediate pleasure more as a distraction from deep-seated emotional pain and self-hatred. It's possible Se is not one of his functions. I can also see a touch of intuition in him that shows up mainly when interacting with or planning to interact with people. It's difficult to say what thinking function he uses.