How do Madeline's appearances in "The Fall of the House of Usher" develop the plot?

Madeline never speaks in the story and only appears as an uncanny, otherworldly figure. In the first sighting, the narrator states that

the lady Madeline (for so was she called) passed slowly through a remote portion of the apartment.

This sighting leaves him with an eery, uneasy feeling.

The second time the narrator sees her is to help Roderick carry her corpse to the vault where she is laid to rest. The third time is when she reappears, alive, and bloodied, only to fall dead on the floor.

The Madeline sightings suggest how sick and close to death the house of Usher is, both the family line and the house itself. Because Madeline is Roderick's doppelganger or twin, her disease and her deathly demeanor are also a reflection of Roderick's diseased self. This is a fatally diseased "house" in both senses of the word.

On a more prosaic plot level, Madeline's fatal disease has exacerbated Roderick's nervous disorder, making him more mentally unstable then ever, and her death at the end leads directly to his. Her death, because it causes Roderick's, leads to the house itself dissolving and falling into the tarn.

First, in order to answer this question, it needs to be determined what the conflict of "The Fall of the House of Usher" is because conflict defines plot since plot is the working out of the main conflict, the complications and the resolution of the conflict. And there is more than one view on what the conflict of this short story by Edgar Allen Poe is.

One view holds that the conflict is (1) rationality versus irrationality, or insanity. In this view, the narrator is rational while Usher is irrational, or insane. The view has difficulties because it cannot explain how everything seemingly "irrational" that was perceived by both the narrator and Usher was proven in the resolution to be actual fact, i.e., Madeline did walk through the door in her funeral shroud; the House of Usher did crumble and disintegrate into the tarn, thus ending along with the family of Usher; the vegetation did prevail.

Another view holds that the conflict is imagination, as defined by the Romantics, versus expected perception. This view of the conflict is supported in the early portion of the story where the narrator talks about "had so worked upon my imagination" and "what must have been a dream." This view is a strong one but doesn't adequately encompass the question of the debate regarding the sentience of vegetation.

A third view holds that the conflict is the appearance of imagination, again according to the Romantic era definition, versus the presence of life, or sentience. This view has a scope that can encompass the other two as follows. Is it rational or irrational to believe in the sentience and intentionality of vegetation? Can imagination (the higher order of reason that creates metaphorical meaning for and order in the world) be trusted to correctly order elements of the world even when perception violates expectation as the narrator experienced upon going to the House of Usher?

If the third view is employed in a discussion of Madeline, then it can be said that Madeline drives and complicates the plot (derived from conflict) of the story by illustrating the conflict of the appearance of imagination (reason) versus the presence of life, or sentience. At the narrator's first glance of her, Madeline was semi-catatonic. She was later fully catatonic and unconscious and lacking any display of imagination, yet she was alive, or sentient as was later proved by her ghastly appearance at the chamber door. Therefore Madeline parallels the vegetable domain, which is similarly "catatonic," "unconscious," and lacking a display of imagination and yet is -- according to Usher and the experience of the House of Usher (both building and people) -- alive, or sentient.

How does Madeline's characterization and appearances in "The Fall of the House of Usher" develop the plot and symbolism? Why is she Roderick’s twin?

Madeline’s presence as a seriously ill person, a dead body in a coffin, and an apparition are key elements in the mood and plot. As the story begins, the reader sees the narrator as a sane commentator on Roderick’s declined state. By the end, the narrator has apparently become convinced of Roderick’s tale and also sees the vision of the dead woman. His claim to have seen a ghost changes the overall tenor of the story: the reader now questions the narrator’s sanity and, by extension, the validity of the entire tale, which now seems impossible rather than improbable.

Because the siblings are twins, the narrator’s initial description of Roderick also largely applies to his sister. He is described as having beauty, delicacy, and fineness, but she is also soft, “wanting moral energy,” and even unforgettable. When Madeline wafts through the room, in contrast, the narrator does not describe her physically: “Madeline … passed slowly through … the apartment, and … disappeared.” In retrospect, we may assume that she was already a ghost. When the narrator sees the dead woman in the coffin, he notes the “striking similitude between the brother and sister….”

The concept of twinness is central to Poe’s story. Roderick and Madeline represent male and female aspects of the same person, and both symbolize the entire family line. Her death, therefore, is not only probably caused by him but is also partly his own. Regardless of which one died first, that death would have begun the destruction of the house of Usher. Once his twin is gone, Roderick knows that his time is limited and no longer cares if the physical house comes down on top of him.