In "The Fall of the House of Usher," how does the house's exterior relate to Roderick, Madeline, and the Usher family?

Without doubt, the setting of Poe's Gothic masterpiece contributes greatly to the atmosphere of isolation, mystery, shadows, decay, and preternatural experiences. That the house is representative of the dying Usher family is indicated in this passage:

...about the whole mansion and domain there hung an atmosphere peculiar to themselves and their immediate vicinity—an atmosphere which had no affinity with the air of heaven, but which had reeked up from the decayed trees, and the grey wall, and the silent tarn. 

• The Usher Family

A family in decline, the Ushers as the remaining descendants are like the mansion that is declining in decay--the narrator speaks of the Ushers as having "molded the[ir] destinities"; they, too, are mysterious, isolated, melancholy, and dying. Both the mansion and the Ushers themselves create in the narrator "a sickening of the heart, a dreariness," and a terror. When the narrator approaches the Usher mansion and as he flees while "the mighty walls rush asunder," it is with these same feelings of terror that he later watches Roderick descend into insanity and be killed by his cataleptic sister Madeline.

• Madeline

When the narrator first sees Madeline, he describes his reaction, a reaction not dissimilar to that of his sighting of the Usher mansion in which he describes his reactions as "a sickening of the heart," "a dreariness of thought" and an unnerving: "I regarded her with utter astonishment not unmingled with dread...." 

Much like the wasting away and preternatural experiences of Madeline and her "similitude" with her brother, the Usher mansion exhibits bizarre sympathies. For example, the deterioration of both Madeline and the house are similar; the door to the vault makes a sharp grating sound similar to the "low moaning cry" with which she falls upon Roderick in her "death agony."

• Roderick

When the narrator first arrives, he senses a atmosphere around the mansion that seems connected with decay and disease. These conditions of decay and disease certainly apply to the Ushers themselves. For instance, Roderick suffers from "nervous agitation," bodily illness, and a "mental disorder which oppressed him." This disorder heightens all his senses to the point that he is tortured by sounds. He paints scenes that are supernatural and horrific, words that can also describe the eventual tearing asunder of the structure of the mansion. Like Roderick, the Usher mansion seems to fall victim to decay. Symbolic of Roderick's break with reality, there is a fissure in the wall of the mansion, a fissure that eventually widens and effects the end of the house, much as Roderick's death comes as he is victimized by his sister's cadaverous attack.

What are some characteristics of Roderick, Madeline, and the house in "The Fall of the House of Usher?"

Interestingly, "The House of Usher" is a double-entendre, meaning both the architectural structure that is the home of Roderick and Madeline and the lineage of the Usher name.  This mansion, once structurally sound, aristocratic, aesthetic, now parallels the family of Usher in its decrepitude.   Because the Ushers have kept their bloodlines too thin, Madeline and Roderick possess, mysterious illnesses; Roderick describes his malady as "a constitutional and a family evil " without remedy. In addition, the Usher line has ben reduced to Roderick and Madeline, who are twins, male and female counterparts of each other; Madeline, too, has a disease that "baffles the skill of the physicians."  Twin that he is, Roderick senses Madeline's condition, and he fears being separated from her as though this separation will deprive him, too, of life.

Madeline dies, Roderick moves her to a vault hoping to preven her being examined in an autopsy, His behavior becomes erratic and bizarre like the fissures in the house. His is a "sensitive nervousness," mirroring the cataleptical condition of his sister in emotion.  In hysteria the ailing Roderick says that his supersenative ears hear Madeline, and she is at the door. With preternatural energy he unbolts a door and in "death agonies" Madeline and he fall dead.  Like the family, the decaying house crumbles.