Is "A Rose for Emily" a short story or novel? What theories can be best applied in that story and why?
This is clearly a short story. People might disagree on how long a work must be to be called a novel, but "A Rose for Emily" is definitely short enough to qualify as a short story.
You might be interested in knowing there is a kind of middle ground, which we call a "novella." This is a work that is too long to be a short story and too short to be a novel. One example is The Old Man and the Sea, by Ernest Hemingway.
According to the Penguin Dictionary of Literary Terms & Literary Theory (Cudden and Preston), there is substantial disagreement, but most scholars classify a work as a novel if it is between 60,000 and 200,000 words. That is not very helpful for most of us, who do not want to count the words in any book, but we are stuck a definition like this because page counts will differ every time a book is published by a different publisher.
The Penguin Dictionary of Literary Terms & Literary Theory notes that there is also disagreement about what constitutes a short story and points out a range from about 800 words to about 30,000 words.
Although the Penguin Dictionary does not give a word count for the novella, it seems that a novella's word count would fall between that of a short story and a novel.
Now, as to theories of the story, those will depend on whom you ask. From a feminist perspective, the theory of the story might be that Rose was unable to become a self-actualized female because of male oppression (her father.) From a socio-economic perspective, we might view this as a tale about how the struggle of a formerly powerful and wealthy family in a changed environment. From a perspective on race relations, we might ask if the change from a slave-based economy and the subsequent fall of Miss Emily's family was responsible for Miss Emily's sad life. Also, we must remember that Faulkner was a very "Southern" writer, and this story is a kind of snapshot of what had happened to the South. There are so many ways to read this story, all valuable and all authentic. I hope you find your own interpretation and value in "A Rose for Emily."