The OP seems (along with a few other posters) to expect a level of certainty that art is incapable of providing. As he points out, even history has such limitations. (Indeed, outside of mathematics, there are few areas of human endeavor in which there is any real degree of certainty.) Like Lamia, I am somewhat taken aback that an English teacher wouldn’t understand this.
Art is an interaction between the artist and the audience. Each brings his/her own experiences to the work and imbues it with meanings that resonate with him/her. These meanings can never be 100% identical between any creator and viewer, or any two viewers, so there can never be any single, unequivocal “true” answer as to the meaning of a given work of art. Why is the Mona Lisa smiling? Is it because that’s precisely how the lady looked that day, or is it because Leonardo is playing a joke on us by painting a secret self portrait? Of course there is no way of knowing. That is the point of art…
Isaac Asimov tells a story (which I recount from my imperfect memory) about being at a science fiction convention where a speaker opined about one of Asimov’s stories. From the back of the room he said, “I don’t think that’s what it means.” The speaker asked, “What makes you think you know what this story means,” to which Asimov replied, “Well, I wrote it.” The speaker responded, “I still say, what makes you think you know what it means?”
It’s a mistake and misunderstanding of the purpose of art to imagine that since there is (and can be) no single unequivocal "answer, " that therefore there is no truth to art. The truth in a work like Macbeth or The Silence of the Lambs is the degree to which the author persuades us that these characters are behaving in ways that are plausible and “true to life.” This does not entail providing answers. In fact, to a certain extent, the more questions a work raises, the better it may be.
Bad fiction can have characters that behave unrealistically, but it can also include stories in which the characters’ motives and personalities are so clear and obvious as to be completely uninteresting. If all you want from a book or movie is to kill a few hours, there is no shortage of crap that will “entertain” without requiring any real thought. In such works, there is probably no point to any deep analysis, because the characters have no depth. As Darren Garrison says, they “act that way because that’s how the author chose to write” them, and that’s all there is. It may be this very characterstic that separates “pulp” from art. Hamlet would certainly not be considered the greatest play in the English language if the title character’s motives were perfectly clear.
So to answer the OP, yes, it is meaningful and useful to think about and discuss the psychology, characteristics, and motives of fictional characters, if those participating in the discussion feel the work is worthy of the effort. As Fretful Porpentine points out, in a classroom setting it teaches students how to analyze and better appreciate works of literature, as well as training them to develop and present their own original perspectives.
Outside the classroom, it is fun and enjoyable to share one’s interpretations with others on Web sites, in book clubs, or in discussions among friends. Sharing your own views, hearing new perspectives from others, comparing them and perhaps revising your outlook; these things greatly add to one’s understanding and enjoyment of a work of art, and indeed can be said to be the real purpose of art.
Perhaps The Silence of the Lambs is, IYHO, on the pulp side of the divide, and doesn’t deserve deep analysis. You have every right to that opinion. (I haven’t read it, although I’ve seen the film.) The people at the Web site you found would seem to feel otherwise.