Does this story relate to William Faulkner's life during the period in which he wrote it?
William Faulkner's story "A Rose for Emily" was first published in April, 1930. Emily's "timeline" begins with her birth during the Civil War (1861-1865) until the early part of the 1930's.
Faulkner is born in 1897, placing his birth at merely twenty years after the southern Reformation ends in 1877. This means that Faulkner's immediate family is part of that reformation time where the South is searching for a new sense of purpose and self-identity.
This is why we find most Faulkner's biographers agreeing in Faulkner's deep interest in Southern history from a very early age, and especially in taking part of the daily lives of the elders of his town in Mississippi, who would sit around the town square to reminisce in times gone by. This is evident in his biography William Faulkner: His Life and Work
...there he sat or stood motionless, quiet, as though held fast by some inner scene or some inner sense of himself
Basically, Faulkner acquires his information from primary sources and, for that reason, he has a qualitative resource that can provide first-hand accounts of what actually happened during the Civil War, during the years before, during, and after slavery and, especially, during the reformation.
This being said, you can certainly see "A Rose for Emily" relating events that Faulkner did not experience first-hand; however, his family, his elder neighbors, and the immediate community was certainly there. So powerful are his sources, that we even question whether Faulkner was actually alive during Emily's "life", and whether he is relating the story of a real woman- one that an entire town, as well as Faulkner, himself, knows personally.
At first we were glad that Miss Emily would have an interest, because the ladies all said, “Of course a Grierson would not think seriously of a Northerner, a day laborer.” [...] They just said, “Poor Emily. Her kinsfolk should come to her.” She had some kin in Alabama; but years ago her father had fallen out with them over the estate of old lady Wyatt, the crazy woman, and there was no communication between the two families. They had not even been represented at the funeral
Furthermore, Faulkner even uses their voices to relate Emily's story. We can definitely agree that what Faulkner tells us about Jefferson County are the stories, experiences, and events that were once related to him from verifiable sources since he was a child.