How essential to "A Rose for Emily" is Faulkner's treatment of tradition and the old south?
Oh, of the utmost. The indictment of the whole town, and the whole South, is evident in Faulkner's continual use of the pronouns "We" and "Our"... It is the warped sense of Southern "duty" that in a strange way is responsible for turning a blind eye on the personal plight of Emily, one of their own.
Futhermore, althought they don't really want to, the force of tradition compels the townspeople to hold Emily in higher regard, simply because her father was a Civil War hero. The compulsion of respect and tradition continues in regard to the "upkeep" of Emily, even though she personally has done little or nothing to deserve any special treatment.
Finally, a Southern sense of the sanctity of privacy will distance the town from one of their own. As a Southerner, I think I can speak for the continuation of this tradition...it's sort of a "don't ask, don't tell," cultural philosophy.
The tradition of "the southern lady" also informs the story. Faulkner calls attention to the lady issue by calling his character "Miss Emily," taking away the "Miss" in the title to release her from the burden that being a lady placed on her. Being the daughter of a gentleman (which gives her title to "lady"), she had less freedom to marry, and she was dominated by her father. Because she was a day, townsfolk needed to "respect" her, which in fact meant not be friends with her. Choosing a northerner and a laborer such as Homer was her way of rebelling against that gender and class distinction that Faulkner associates with the old south.