I only see a couple of references (or at least direct references) to the Civil War in this story. The first comes right at the beginning of the story when the narrator makes reference to the Union and Confederate dead in the cemetery. The second refers to the old soldiers who come to pay their respects at Miss Emily's funeral.
I think that these mentions of the war are made to emphasize how Miss Emily is part of the ancient past. She is associated with the Civil War, even though she herself was not old enough to have been the contemporary of the men who fought in it. The references make her appear to be part of a completely different time than the society in which she actually lived.
Just a quick note: "A Rose for Emily" is obviously fiction. When it was published has nothing to do with when Emily was born. You can't equate the story's publication date with Emily's death. And how long Faulkner was working on the story is irrelevant. I can write a story about someone born in 1508, or 1203, or 2098. The date I publish the story has nothing to do with my character's age. There is no reason to assume that Faulkner wrote the story immediately after Emily dies, unless there is some external evidence that I don't know about. You can figure out her chronology from the story if you want, but you can't figure it out by assuming the publication date has any chronological relation whatsoever with Emily's death.
The Civil War mentions are important to the story because Faulkner is writing about the reconstruction stage of the American South following the war. Notice that everyone is poor. The war devastated the South. It not only destroyed the South's economy, but most of the battles, etc., took place in the South. The region was ravished by the war.
Actually, Miss Emily Grierson was probably born either before or during the American Civil War. ("A Rose for Emily" was first published in 1930, and Faulkner had apparently been working on it for several years. Since Miss Emily was around 70 years old, this would put her fictional birth between 1850-1860.) She would have been one of the last survivors of this epic conflict, so both she and the war serve as a reminder to the glory days of the South. Well into the 20th century, the War Between the States served as a dividing line between various social and ethnic beliefs. The beaten Southern states took many decades to recover from the economic and political restraints they were forced to endure during Reconstruction. Once they had earned an equal place in national politics, many Southern leaders--a number of whom were important members of the Confederate military and state governments--sought to build a New South that reflected an image of the CSA in its heyday. By the turn of the century, however, with most of these men dead or dying, the South was changing once again; the old ante-bellum ways were being disregarded for a more modern approach. Miss Emily remained a symbol of the old ways; the younger citizens of Jefferson considered her a relic of a time best forgotten. Her own peculiarities only gave more ammunition to those who sought an end to the old ways that were now considered decadent in the first decades of the 20th century.