Was the title chosen before or after the story was written?If you know of a source that supports your answer, can you please provide it?
Faulkner did not specifically tell us whether he chose the title before or after completing the story, but the story's repeated rose imagery, its themes, and the underlying meaning contained therein in all hint at an overall plan/structure based on repetition of the "rose" concept. This implies that Faulkner chose the title early on. There are different uses of "rose" in the text. For instance, the word is often used as a verb. The color rose is also apparent -- specifically, in what would have been Emily's marital bedroom. The "Miss" of Emily's name is missing from the title, and this helps build one of the major themes of the story. Emily's spinster life is the result of missed chances/poor choices (some under her control, and some out of her control). "Miss" Emily is a spinster; "Emily" is a girl, young and available for romance (the rose). This missed chance of real romance is driven home by the twisted approximation of the story's climax: unable to experience the happiness/romance of a real relationship, Emily sleeps next to a corpse.
Faulkner has been quoted as saying that the story's title is meant to represent a gesture -- that is, expressing kindness (giving a rose) to a forgotten, downtrodden woman (Google Books).