I agree with number 4. If you look in most major American literature anthologies, Poe is categorized with the Romantics, mainly because of his Gothic tendencies.
While some of Poe's detective stories are closer to parts of Naturalist philosophy (logical reasoning, fatalism, etc.), they still significantly rely on Romantic elements.
Neither. Poe was a classic example of a Romantic, even more European than the English Romantics. His works contained substantial strains of the Gothic and the fantastic, which takes them away from the point of realism on the continuum. His emphasis on the visceral could be called naturalistic, but his works were too far removed from naturalism to fit into that genre.
Poe liked to explore the darker sides of the human psyche. He was not afraid of delving into the more morbid thoughts and fantasies of man. As mentioned above, he also wrote many love poems. He loved several women in his life and found little joy with any of these women. For the reasons above I would say Poe was more a realist. Realist lovers may not agree but there are several stories that fit the genre.
"Edgar Allan Poe's mystery and science fiction are underacknowledged by realist literary critics. Poe was a pioneer in both genres, and together they constitute, in bulk, half of his short tales. However, the favorite Poe works among realists include 'Ligeia', 'The Fall of the House of Usher', and 'William Wilson'. These are the Poe works that are closest to conventional realistic fiction: there is an emphasis in these works on psychological portraiture, and the study of human relationships. This is quite common, to emphasize those works in an author's canon that correspond to the conventions of conventional literary thought, and ignore the rest. There is no mystery in these works, and the fantasy, where it exists, is strictly supernatural, with no scientific overtones."
Poe is typically considered one of the premier writiers of the American Romantics, and for that reason probably falls closer on the continuum to realists than to naturalists. Realists tended to focus on the actual state of existence, while naturalists were typically concerned with the scientific laws that underwrote man's existence. The Romantics teneded to be more concerned with the individual's experience, and less concerned with structural literary issues such as form and genre. While Poe is probably best known for his horror stories, with their macabre plots and generally less-than-pleasant endings, his Romantic leanings can be seen in such poignant verses as "Annabel Lee," a piece that in both structure and spirit is about as romantic as a poem can get.