Why should Hester be allowed to take the Scarlet Letter off?

Hester Prynne was made to wear the scarlet letter on her chest as a punishment for commiting adultery, which she had to wear till death. She retired into wilderness along with her daughter Pearl. She suffered her humiliation with an iron heart. She suffered for the sins, not only of her but also of the co-sinner about whom she didn't confess, like the Christ who suffered for the sins of whole humanity. Being skilled seamstress she earned a secure and dignified income. Day by day, year by year, her penance made her a saintly figure. She became famous for her piety. She helped everyone who came to her doors.She is even termed by Hawthorne as a self-ordained "sister of mercy". With such a transformation of her grace the scarlet letter on her chest had lost its significance and it shone like a cross, therefore there was no need for it and it was better to allow her to take off the scarlet letter.

Hester Prynne was originally forced to wear the scarlet letter as a form of punishment, which distinguished her as a sinner from the rest of Salem's devout citizens. Seven years after standing on the scaffold in front of her neighbors to be publicly ridiculed, shamed, and criticized, Hester becomes an integral part of her community. Hester is a talented seamstress, who makes luxurious items for the citizens of Salem and becomes a selfless, volunteer throughout her community. The citizens begin to accept and revere Hester for her benevolence and compassion. She aids the sick, sympathizes with those in need, and offers help to the desolate community members. In chapter 13, Hawthorne refers to Hester as a self-ordained "Sister of Mercy." The community even begins to associated Hester's scarlet letter with the word "Abel" because of her strength and tender heart. Chillingworth even mentions that there are discussions among the ministers and officials of the community about removing the letter. Overall, the scarlet letter loses its original sting and Hester becomes a respected, valued citizen. She has paid the price of her sin by enduring ignominy for seven years and has become a positive influence on her community. 

How does synesthesia play a part in understanding and performing music?

The idea behind synesthesia is that, basically, one sensory input generates multiple responses. Think of it as, when you hear a note, you might also "see", or perceive, a color related to it. It's not limited exclusively to notes generating colors; one famous example is Daniel Tammett, who was able to recite a very large number of the digits of pi from memory by associating numbers with colors and using the memory of those colors instead of trying to remember the digits.
This phenomenon gives you multiple ways to construct music; instead of just stringing together a sequence of notes and sounds, you can effectively paint a picture or tapestry of color. This also aids, particularly, in note identification. We may have trouble normally distinguishing between two adjacent notes, but it's far easier for the eye to distinguish between two similar shades (say, blue and indigo). So if you hear two notes that evoke different colors, you can better identify specifics.
Synesthesia, basically, gives a greater toolbox to work with. If hearing notes evokes specific stimuli, you can use those stimuli to aid in picking apart music and constructing it yourself. It should be noted that, among people with synesthesia, they tend to have "perfect pitch" at a much higher incidence, which is also linked to musical aptitude.
One final note: synesthesia's effects are not the same among all people with the phenomenon. Someone might see a note as blue that another person sees as fluorescent yellow.
https://science.howstuffworks.com/life/inside-the-mind/emotions/synesthesia1.htm

Where does Stanley live?

Stanley, or Stanley Yelnats IV, to give him his full name, lives in California. As he tells us himself, his great-grandfather—the first Stanley Yelnats—lost his fortune on his way out west from New York. The stagecoach he was riding in was robbed by a notorious outlaw by the name of Kissin' Kate Barlow. The present-day Stanley thinks that his great-grandfather's robbery was the catalyst for a streak of bad luck in the Yelnats family that has yet to run out all these years later. Furthermore, he also thinks that if the first Stanley Yelnats hadn't lost all that money he'd made on the stock market, then he'd now be living in a mansion on a California beach instead of being crammed inside a tiny apartment that smells of burning rubber and foot odor.

Compare and contrast the relationship between Jem and Scout with the relationship between Atticus and Aunt Alexandra.

Both Jem and Scout and Atticus and Alexandra have fairly typical sibling relationships throughout the novel. In both sibling groups, each person cares about their sibling's well-being and disagrees with their siblings on occasion. Similar to Atticus and Alexandra, Jem and Scout are prone to arguing with each other, and both siblings struggle for influence and control. Like Atticus and Alexandra, Jem and Scout spend a significant amount of time around one another and attempt to console each other during difficult times. Alexandra offers to help Atticus raise Scout, and Jem demonstrates affection for Scout by walking her to the Halloween festival, buying her a baton, and attempting to prevent her from arguing with Alexandra.
Unlike Atticus and Alexandra, Jem and Scout do not have fundamentally different ideologies regarding race, heritage, and education. Jem and Scout also express their affection towards one another more than Atticus and his sister. Scout also looks up to her brother and follows in his footsteps, which is not the case in Atticus and Alexandra's relationship.

In what way does Emilia’s role develop in Othello?

At the start of the play, Emilia appears to be subservient and manipulated by her husband, Iago, but by the end she has gained in self-confidence and has seen him for the villain he is; finally, she denounces him and reveals his evil to the world. Initially, Emilia does what Iago bids her, despite his lack of warmth to her. Although she feels guilty, she steals Desdemona’s handkerchief for Iago to use as evidence of Desdemona’s infidelity. The audience can understand that Emilia simply wants to please Iago in doing this, saying "I nothing but to please his fantasy," which she quickly regrets when he fails to praise her and she feels remorseful about deceiving her friend Desdemona. It is this friendship that seems to move Emilia to find her moral compass, even though she becomes bitter about her marriage.
 By the end of the play, the submissive Emilia criticizes men in a way that would have been seen as powerful and almost heretical to the Jacobean audience of Shakespeare’s time.  She says "But I do think it is their husbands’ faults/If wives do fall," thus blaming men for leading women astray, which would be unusually opinionated for a woman in the male-dominated patriarchal society of the time. It is ironic that Iago underestimates Emilia, as she finally exposes him for being the cause of so many deaths. Iago has made a mistake based on his own lack of morality and assumptions of others; he never thought that Emilia would betray him and sacrifice herself to "love and duty," which is precisely what she does.

What are common themes in Frankenstein by Mary Shelley and The Kite Runner by Khaled Hosseini?

Arguably, the greatest theme of Frankenstein is the idea of secrecy. Victor’s passion to create life is shrouded in secrecy. His success is also shrouded in the same secrecy. When the monster comes to life, Victor cannot confess his creation to anyone, due to his fear and shame. So he keeps the truth inside. When the monster starts killing, he has to keep the secret even tighter. Finally, Victor reaches a point where he can confess, and he does so to Walton.
The Kite Runner plays with the same idea of secrecy after Amir's childhood betrayal of Hassan.
As Hassan is running to fetch a kite, a number of boys surround him: Wali, Kamal, and Assef. Assef ends up raping him. Amir sees this act, but he acts like he did not, because he was afraid to stand up to the boys. Guilt racks Amir. Compounding this guilt is the fact that Amir frames Hassan and gets him in trouble with Baba. Hassan then moves away. Amir holds this secret inside for years, and it haunts him as an adult.
Along with this idea of secrecy, there is the weight of guilt for both Amir and Hasan. If you think about this a bit, you will see there is a lot of fruitful content.

Both Frankenstein and The Kite Runner deal with several similar themes, though separated by years of history and with very different authorship. In Frankenstein, Victor Frankenstein is appalled by the monstrosity of his creation. He is devastated by what he has done, and spends the entire novel trying to kill the monster and avenge the deaths of his loved ones. 
Similarly, Amir spends much of The Kite Runner trying to escape the monstrosity he witnessed when he saw Assef beat and rape Hassan. Amir tries to make his own amends by adopting Hassan's son. 
Father-son relationships are also themes in both novels. Though Victor's relationship with his father in Frankenstein is fairly typical, his relationship to his created son, the Creature, is atypical. He rejects his "son" entirely, and the Creature is left alone and parentless. The Creature becomes monstrous because he is never shown love. Victor never has biological children because the Creature murders his wife on their wedding night. 
In The Kite Runner, Amir's father is ashamed of his introverted and bookish tendencies, which creates tension in their relationship, as does their move to the US. Like Victor, Amir is unable to have children with his wife, and instead adopts his brother's son in an attempt to create a family. 
Because both novels deal intimately with shame, making amends, and family relationships, the theme of identity can be said to encompass all these smaller themes into a larger, unifying theme that runs across both stories.

How is the physical journey in The Great Gatsby an important element and how does it add meaning to the story?

The only character who I think makes any significant "physical journey" is Jay Gatsby. We learn later in the novel, after his death, that he grew up poor in the Midwest. Through hard work and exceptional discipline, however, Gatsby saved money, used his experience as a soldier in World War I to study at Oxford, then returned to the United States and worked for Meyer Wolfsheim as a bootlegger in New York.
Gatsby's physical trajectory is similar to that of Fitzgerald, who came from the Midwest (St. Paul, Minnesota), entered World War I as a second lieutenant, then returned to America to write and live in New York. Some of his other characters perform a similar physical journey. Dexter Green, the protagonist of "Winter Dreams," comes from small-town Minnesota, goes to an Ivy League university in the East, returns to Minnesota and enters the laundry business (a successful venture that is briefly interrupted by the war), then moves to New York to expand his wealth.
The importance of these physical journeys is that they are key to the characters' attainment of the American Dream—that is, the journey is what allows them to become upwardly mobile, though Gatsby's path to upward mobility (i.e., bootlegging) is illegitimate and becomes a way through which Tom Buchanan shows Daisy that Gatsby is unsuitable for her. In this way, Fitzgerald exposes the hypocrisy of the upper class, which partakes in drinking the liquor that was illegal at the time (Buchanan drinks liquor while with his mistress Myrtle) but thinks it improper that Gatsby helps to sell it.
Gatsby's physical journey could also be read as key to his downfall. Fitzgerald's work is hopeful about the American Dream but skeptical of its promises. He wants to believe that smart and ambitious people like Gatsby can succeed and belong among the wealthy class but is skeptical that they can be fully accepted.