Your original question contained more than one question, so I have edited it down to focus on the exposition of this great short story by Truman Capote. Hopefully this will help you to go and complete the rest of your plot chart and identify the other stages of this tale.
Remember that the exposition or basic situation of a story is the opening of the story when the characters and their conflict are introduced. So, looking at the tale, it is the first few paragraphs that introduce the setting:
Imagine a morning in late November. A coming of winter morning more than twenty years ago. Consider the kitchen of a spreading old house in a country town. A great black stove is its main feature; but there is also a big round table and a fireplace with two rocking chairs placed in front of it. Just today the fireplace commenced its seasonal roar.
The coming of winter and the cold exterior matched by the warm interior thanks to the fireplace is described as we are plunged into a sprawling country house. The two central characters are then introduced: the "woman with shorn white hair" and the seven year old boy, who have a very special relationship, in spite of the age difference between them: "We are each other's best friend."
The conflict becomes clear by the somewhat mysterious first words of the woman as she says "It's fruit-cake weather!" Having thirty cakes to bake introduces the conflict as the characters have to fight against poverty to enable them to buy, make and post all of the fruit-cakes for Christmas presents.
Hopefully this will help you identify the rising action as they struggle to get all the ingredients for the cakes, the climax as they manage to complete them and celebrate Christmas, and then the falling action as the narrator tells us how time moved on and his special friend died.