When writing dialogue for a short story or novel, I should always switch out tags like "said" or "says" with more exciting ones like "whispered", "guffawed", or "screeched", right?
Not always. Often times, "said" or "says" allows the emphasis in a story to be put on the dialogue instead of the on the way that the character said it. If it is necessary to the story that your character is whispering, then it is appropriate, but there is nothing wrong with using "said".
In "The Great Gatsby", what is the significance of the green light that Gatsby sees at the end of Daisy's dock?
The green light, for Gatsby, is something he holds on to while he is waiting for Daisy to come to a party. In the novel however, it has more meaning than that, which is given to the reader towards the end of the novel. Nick contemplates what the new world must have meant to the Dutch sailors who had come to America. He pictures them seeing his green hills in the distance, which is where we are given the connection to Gatsby's light. It symbolizes the hope and the purity of the American dream, and is manifest in Gatsby's life.
It seems like teachers are always trying to pull out symbolism and meaning from ordinary details and minor passages in novels or poems. Why can't something be the way it is just because the author felt like it? Does all of it really have meaning?
In literary fiction, which is mostly what is studied in school, each detail does have meaning. Teachers spend extra amounts of time focusing on the little details that point to a bigger effect, like rain making the day feel sad or the color yellow meaning happy, because these create subconscious connections between the author's intent and the impact on the reader. These are called archetypes. When reading along for pleasure, these details subtly create a larger picture, but when we are learning the importance of this technique, there is an extra emphasis put on it to help us understand.