When Melba was born, her scalp was injured and she got an infection. Why didn't her health improve immediately?

Melba's health didn't improve immediately because the white nurses neglected to continue with the necessary treatment that would have cured Melba of her infection.
When Melba was born, the doctor had injured her scalp with the forceps. As a result of this, Melba later incurred a massive infection. Accordingly, the doctor had ordered an Epsom salts and warm water treatment every two or three hours, but the white nurses had neglected to be diligent about continuing the treatment. Melba's mother only found out about the treatment by chance, when a black janitor mentioned what he heard the doctor tell a nurse in the operating room. Accordingly, the nurses were supposed to use an Epsom salts and warm water concoction to irrigate Melba's head.
After she confirmed this fact with a nurse, Melba's mother then continued the treatment herself. After two days, Melba's health began to improve, and she eventually made a full recovery.

How did Greece’s geography affect recent world events?

The question asks how Greece’s geography has affected recent world events. The term “recent world events” is rather broad, so let’s narrow it to the issue of refugees from the conflict in Syria. The fundamental elements of Greece’s geography relevant to this issue are:
1. Control over numerous small islands in the eastern Mediterranean Sea.
2. A relatively rugged, mountainous border region with both Turkey to the east and the rest of the Balkans to the north.
3. Membership in the European Union.
These three factors have conspired to make Greece the primary route through which refugees from the Syrian conflict have attempted to escape into Western Europe and seek asylum (and ultimately permanent residence) there. Many Greek islands are close enough to Syria and Turkey to make it tempting for refugees to attempt to navigate to them in various small craft. The rugged border with Turkey makes infiltration possible through that route. Finally, EU membership makes doing so attractive, because the free travel allowed among EU member states has made control of refugees problematic. Additionally, standards of living in these states are among the highest in the world, so there is a belief among refugees that “if they can just get there,” economic opportunity and freedom will be waiting. If Greece were not part of the union, this latter impetus might well not be present, as Greece itself is struggling economically and does not have a large economy which could absorb substantial numbers of refugees.
https://data2.unhcr.org/en/situations/mediterranean?id=83

What are equinox and solstices dates? What happens to light and dark during these times? Where is it dark or light?

Equinoxes and solstices happen due to the tilt of the Earth on its axis.  Solstices occur when the Earth is at a maximum angle away from the Sun.  During the summer solstice the sun is directly over the Tropic of Cancer.  North of this latitude line locations experience the longest day of the year (meaning they get the most sunlight on that day).  The summer solstice typically occurs around June 21st.  Around December 21st, we experience the winter solstice.  The winter solstice occurs when the sun is directly over the Tropic of Capricorn.  The winter solstice represents the day with the least amount of sunlight in the Northern Hemisphere. 
Equinoxes, on the other hand, represent the point when the Earth’s axis is relatively straight in comparison to the sun.  During these days, the sun shines directly on the equator, causing an almost equal amount of daylight and night time.  The vernal equinox occurs on approximately March 20th and the autumnal equinox happens around September 22nd. 
Please refer to the National Weather Service website link below, which provides a visual explanation of this concept. 

What is the meaning of the phrase "as if to a vigil beside a perfectly tangible body" in the first chapter of The Great Gatsby?

A vigil beside a body would be a funeral service (actually a vigil is an all-night prayer service) held in the room where the dead body is laid out. 
So, the manner in which Tom and Miss Baker strolled back into the library suggested they were going to a funeral.  They were solemn, sad, and probably awkward, not meeting each other's eyes.  The awkwardness is very common at funerals, and it is suggested by the fact that they had "several feet of twilight between them," and by the way Nick felt at supper a few minutes before: "wanting to look squarely at everyone, and yet to avoid all eyes."
Of course, Tom and Miss Baker were not literally about to step into a room with a dead body.  This is figurative.  The author, knowing it is a figure of speech, adds the words "perfectly tangible" to make it more vivid: it might be a figure of speech, but their behavior was exactly as if they were going in to view a real body.  This is similar to when people add the word "literally" to heighten the intensity of a figure of speech, e.g., "I was literally dead on my feet."
The awkwardness and sadness was brought on by the fact of Tom's mistress calling the house during supper, which made it obvious to everyone present that Tom and Daisy's marriage was miserable. 

How does Boyne present the differences between Bruno's life in Berlin and "Out-With"?

Boyne, the author of The Boy in the Striped Pajamas, uses simple and direct language that echoes the words and tone a 9-year-old boy like Bruno might use to describe his life. Boyne's diction, or word choice, is very straightforward; it emphasizes the setting and the impressions that such a setting makes on the characters involved.
In Berlin, Bruno's family has a butler, a maid, and a cook. He attends school, and he lives relatively close to his friends. Bruno's family home is large; in fact, at the beginning of the second chapter, the house is described as "enormous." Bruno even has his own bathroom, and he is yet to have discovered all the "nooks and crannies that he hadn't fully finished exploring" in this home. His neighborhood is full of other large houses. His house is "on a quiet street and alongside it were a handful of other big houses like his own, and it was always nice to look at them because they were almost the same as his house but not quite, and other boys lived in them who he played with (if they were friends) or steered clear of (if they were trouble)." Lively market stalls "piled high" with fresh fruits and vegetables amused Bruno and lent a comforting air of activity to the area. These descriptions lead the reader to believe that Bruno led a comfortable life in Berlin.
Upon Bruno's arrival in Out-With with his family, he sees that "everything about it seemed to be the exact opposite of their old home and he couldn't believe that they were really going to live there. . . . When he closed his eyes, everything around him just felt empty and cold, as if he was in the loneliest place in the world. The middle of nowhere." This bleak introduction to Out-With contrasts powerfully with Bruno's old life in Berlin. Within moments of arriving to his new home, Out-With, Bruno experiences a terrible stomachache, and he wants to cry; these emotional reactions to Bruno's new home only foreshadow the pain to come.

In chapter 16 Widge has the opportunity to observe Mr. Shakespeare. What does he think of him?

Widge thinks that Mr. Shakespeare has an ill-tempered disposition and a tendency to be difficult. 
In Chapter 16, Widge has an opportunity to observe Mr. Shakespeare while the latter is getting ready to play the part of the ghost in Hamlet. Both Widge and Mr. Shakespeare are in the tiring-room (or the theater's dressing room). Mr. Shakespeare is dressed in armor and "touching up his ghostly white makeup." The effect is startling to Widge, who can't help staring at Mr. Shakespeare in the mirror. 
After Mr. Shakespeare puts the finishing touches on his makeup, he proceeds to stare into the mirror. As far as Widge can tell, Mr. Shakespeare doesn't appear to be focusing on his appearance or anything in particular. Instead, the playwright's faraway expression hints that he is focused on other matters. Widge speculates that Mr. Shakespeare may be thinking about his next play. 
Mr. Shakespeare soon catches Widge staring at him, and he verbally reprimands him. Widge is embarrassed when Mr. Shakespeare questions his ability to do his job well. Later, Widge discovers that Mr. Shakespeare is prone to melancholy and impatience.

Why would a small state be more powerful if representation was numerically equal per state?

The issue of representation in the U.S. Congress was hotly debated at the Constitutional Convention in 1787. The Convention was originally called to revise the Articles of Confederation, which had been the government of the original 13 colonies of the United States during and after the American Revolution. Leaders such as Benjamin Franklin feared the government under the Articles was too weak, and so they met to revise, and ultimately replace, the Articles with a stronger U.S. government, creating the Constitution.
One of the biggest issues that emerged in creating a new government was state representation in the national Congress. Large states supported the Virginia Plan, which advocated replacing the Articles with a stronger national government and basing a state's representation in Congress on the population of that state. Smaller states, such as Rhode Island and New Jersey, opposed this plan because it gave more representation, and therefore more power, to larger states. The higher a state's population, the more representatives they would be able to send to Congress, and thus the more votes they would have in Congress.
Smaller states therefore preferred the New Jersey Plan, which called for equal representation in Congress—every state would get the same number of representatives in Congress. This plan would mean that a smaller, less populous state, such as Delaware, would have the same number of votes, and therefore the same influence, as a larger state like New York or Virginia.
In the end, the Great Compromise helped settle the dispute and created the bicameral (two house) Congress we have today. Representation in the lower chamber of Congress, known as the House of Representatives, is based on each state's population. The upper chamber, called the Senate, allows every state to send the same number of representatives: 2 senators per state for a total of 100 senators.