Is Emily insane?
I would think that this would depend on the definition of "insane" that you are using when you answer this question. I think that you can argue it either way depending on how you define the term.
On the one hand she clearly does not behave in any sort of a normal way. She denies that her father is dead for a few days before she finally allows him to be taken away. Much worse, she seems to have been sleeping in the same bed as Homer Barron's remains for decades. Surely someone who kills a person and then sleeps in the bed with their corpse is not completely sane.
But, at the same time, she is clearly able to function and to know what society will think of her (to some extent). She makes sure that people do not know what she has done (by, for example, not telling the druggist why she wants arsenic). She teaches art for a while. She rebuffs the city leaders when they try to get her to pay taxes. She is clearly not insane in the sense of being unable to function in her society.
Please see the link below for more answers to this question.