In "The Storm" by Kate Chopin, what circumstances from Part I significantly affect the story's events?

As far as the plot is concerned, the introduction of the treacherous storm in Part I explains why Bobinot and Bibi are marooned at Friedheimer's store for the afternoon. This stay keeps them away from Calixta, whom they assume is eagerly awaiting them at home.

While they wait out the storm, Calixta engages in a passionate afternoon tryst with Alcee Laballiere, who seeks shelter from the storm in Calixta's home.

While readers would expect an extramarital affair to tear a family apart, both Calixta and Alcee's families are content, and even strengthened, after the storm. The last line of the story reads, "...the storm passed and everyone was happy."

Also introduced in Part I is Bobinot's naivety concerning the thoughts and feelings of his wife. As he and Bibi marvel at the storm, Bobinot says to Bibi, "Mama'll be 'fraid, yes." However, we discover in Part II that Calixta barely notices the storm nor the absence of her husband and son. Bobinot does not know his wife's heart, which likely explains Calixta's desire to seek fulfillment in someone else, albeit temporarily.

Chopin's concise, evocative technique is evident in Part 1 of "The Storm." With a few strokes of the pen, Chopin establishes the character profile and relationships of the key characters: Bobint is a compliant, if somewhat dim, husband and affectionate father; Bibi is remarkably observant, wise for his years; neither character is truly "afraid" for Calixta, whom they both clearly love. The impending storm is a classic example of foreshadowing; it symbolizes the turbulence of Calixta's repressed desire and the threat to her marriage; it also serves efficient double duty as the element of the plot that will delay father and son, creating the opportunity for her affair. Yet even in the story's opening, we are assured that this is a passing storm, literally and figuratively; neither father nor son is substantially threatened by it. Bobint buys the can of shrimps his wife prefers and "stolidly" waits out the storm, confident of his dinner, and ultimately, his wife.

In "The Storm," Calixta has an affair with another man while her husband and son are waiting out a storm at a local store. In this story, the affair is not presented as something evil or even something that would make Calixta feel guilty. Rather, the affair is presented as a celebration of women's sexuality. It is a celebration of the freedom to express female sexuality by Chopin herself. So, as it is an expression of passion, note how the storm is analogous to Chopin's expression of female sexuality as well as Calixta's passion within the context of the story. 

. . . the storm burst. It shook the wooden store and seemed to be ripping the furrows in the distant field. Bibi laid his little hand on his father's knee and was not afraid. 

The clouds roll in with their "sinister intention from the west, accompanied by a sullen, threatening roar." The oncoming storm foreshadows something apparently "sinister." The storm shakes the store while Bibi and his father wait. The release of the storm parallels the release of passion between Calixta and Alcee. 

The storm seems threatening and violent when it begins. With the analogous affair, Calixta's act could threaten her marriage. However, in the end her marriage stays intact. Chopin shows that just as men have moments of passion and lust, so do women. If it is healthy for men to have urges, it is healthy for women as well. Likewise, the storm is nature's way of releasing energy. It was simply a temporary surge of energy (storm and the affair). The storm was foreseen as threatening but was just a passing storm. The affair has the potential to threaten the marriage, but it was just a passing moment of passion. Thus, the story ends with "So the storm passed and every one was happy." 

In "The Storm" by Kate Chopin, how does the setting influence the plot, character actions, emotional states, and motivations? What role does weather play?

The setting, especially the storm itself, is symbolic. Everything seems calm, knowable, before the storm, but then as the storm develops and grows more intense, so do the feelings of Calixta and Alcée inside the house. When lightning strikes, "Calixta put her hands to her eyes, and with a cry, staggered backward. Alcée's arm encircled her, and for an instant he drew her close and spasmodically to him." It is a key force, certainly, because it brings these two characters together, alone, as they have not been since Calixta's marriage.

Further, the storm not only brings them together into the same physical space, but it also reignites the long-dormant passion between Alcée and Calixta. As the storm gathers,

[Alcée] pushed [Calixta's] hair back from her face that was warm and steaming. Her lips were as red and moist as pomegranate seed. Her white neck and a glimpse of her full, firm bosom disturbed him powerfully. As she glanced up at him the fear in her liquid blue eyes had given place to a drowsy gleam that unconsciously betrayed a sensuous desire. He looked down into her eyes and there was nothing for him to do but to gather her lips in a kiss.

The storm initiates Calixta's fear for her husband and son, and this heightened emotional state shifts quickly into another intense emotion: desire. The intensity and fervor of the storm matches the emotions of these characters. While it rages outside, dropping "crashing torrents" of water, inside, Alcée marvels over the "generous abundance of [Calixta's] passion, without guile or trickery, [which] was like a white flame [...]." While the literal water rages outside, a figurative fire burns inside. The storm both begins and then matches their passion and fervor.

"The Storm" by Kate Chopin represents the nucleus of the story. Without the storm, the adultery would not have happened.  Chopin knew how to draw the reader into the upcoming storm both in the weather and in the major character's life.  In the story, there is the before the storm, during the storm, and after the storm which essentially provides the plot of the story. 

The weather enables the entire story to progress.  Calixta's husband and child are caught in the weather and cannot come home.  Alcee probably would not have ridden up to find shelter without the fierce storm on the way. 

To add to the romantic interlude between the lovers,  as the storm increases so does their passionate love making. 

They did not heed the crashing torrents, and the roar of the elements made her laugh as she lay in his arms. 

When the storm ended, Alcee rides away.  Soon, her family comes home, and Calixta does everything she can to please them.  Alcee writes to his wife and tells her to stay longer on her visit, which pleases the wife.

After the storm, Chopin tells the reader that everyone is happy. 

http://www.enotes.com/storm-kate-chopin-salem

I think that one of the clues that the setting is important to the story is the title.  Titles usually have a strong thematic connection to the story.  I also like to consider the information given in the first sentence of the story.

The leaves were so still that even Bibi thought it was going to rain.

This sentence also refers to the weather, because it mentions “leaves” and “rain” and it also already begins to characterize.  We know that Bibi is a child, and the change in weather is perceptible even to a child.  Consider how the storm is described, as having “clouds that were rolling with sinister intention from the west, accompanied by a sullen, threatening roar.”

Already, the storm is described in metaphors, so we begin to get the idea that the storm is a metaphor for the story, and deeply important to the plot and meaning.

Read the enotes summary of "The Storm" here.