Roderick and Madeleine, as twins, are part of an enigmatic and not entirely understood group of individuals. Twin siblings have demonstrated through research that they share a tremendous psychological connection, and some even dare say that this connection transcends the physical world.
Evidence of this is found in the story in the words of the narrator:
A striking similitude between the brother and sister now first arrested my attention; and Usher, divining, perhaps, my thoughts, murmured out some few words from which I learned that the deceased and himself had been twins, and that sympathies of a scarcely intelligible nature had always existed between them.
In their case, Roderick can sense the feelings and emotions of Madeline, who is in a catatonic state for reasons that we can only speculate. Yet Roderick feels that Madeline "walks about." Since they are indeed twins it makes it seem almost possible that the two, sharing such a rare genetic uniqueness, are also able to establish communication telepathically.
Therefore, the twin connection is a clever way to add enigma to the character of Roderick. That the twins are born to a family with a genetic predisposition to self-destruction makes the problem of the story all the more tragic.
The twin relationship enhances the Gothic effect of the grotesque: Here are a brother and sister who have a bizarre relationship--there blood line is too thin; their being twins thins the line even more. Roderick suffers from a morbid nervous condition while his sister has a strange apathetic illness with catalepsy that keeps Roderick nearby in the mansion that decays as the family does. And, since Madeline is his own flesh and blood, Roderick has the twin intuition that senses what the other feels and thinks. Thus, when Roderick tells the narrator that Madeline still moves in the house and is not dead, there is more credibility given to his declarations. In the final horror, Madeline "bore him to the floor a corpse"; they are united in death and they were in birth--an eerie ending made more plausible because they are twins.
In the 19th century, there was a lot of specualtion about the supposed psychological link between twins, and Poe capitalizes on this in "The Fall of the House of Usher". The fact that this is the case for Roderick and Madeline adds to the horror when Madeline is buried alive. The "psychic link" between them is what creates the paranoia in Roderick and the eventual realization about what he did to his sister. He seems to sense her fears and her anguish, and they are reflected in his behavior. When she finally does appear outside the door, Roderick is certain that she is there before the door is even opened. Again, this adds to the horror of the tale. Having the two die nearly simultaneously reflects the link between twins.
I believe that Poe had the two siblings be twins, rather than just brother and sister, because he wanted to emphasize just how close they were. People often say that twins are closer than regular siblings because they are the same age and have grown up together from the moment they were conceived.
The siblings' closeness is relevant because of how closely their fates are connected. They both are seemingly quite ill and eventually, they die at the same moment -- their fates are intertwined. By writing them as twins, Poe makes it more sensible for them to have such a bond that they would share a tragic fate.
Yes. There are several. First, traditionally twins are often joined or linked in some way. This makes it more likely that they would stay "in contact" after her death. Second, twins are a kind of double; if one is dead, the other is too. Third, this allows them to function as the male and female principle in a psyche, with the house being the larger psyche. That way, their connection is like a link to the subconscious…and maybe madness.