In "The Garden Party," why was the main character undeveloped?

Katherine Mansfield leaves the character of Laura Sheridan relatively undeveloped because writers of this period, Modernism, were experimenting with a new, different style. World War I had recently ended, and the war had produced a very destabilizing effect on modern society.  Modernist writers wanted to capture that destabilization in their work, so many were leaving out expositions, character development, and tidy resolutions at the end of the story.  
Modernist writers wanted their readers to become their partners in creating meaning.  What Laura learned about separations between social classes, reality and illusion, and life and death is left unexplained.  Through observing what Laura saw and heard in the wake of her neighbor's death, Mansfield wants readers to draw their own conclusions about these essential conflicts, rather than using Laura to deliver a conclusive message.  Laura's inability to express herself with regard to what she saw at the Scott's house mirrors the inability of modern people to express what they felt about the experience of World War I and the uncertainties of modern life.

Why does the postmaster not have his own name in the story "The Postmaster" by Rabindranath Tagore?

To answer this question, we will have to understand Tagore's perspective on his beloved India. To him, India had long been mired in superstitious traditions, and he argued for a more open approach towards science and global influences. Tagore also deplored in his lifetime the constant conflicts between opposing factions of Muslims and Hindus; as an enlightened thinker, he argued for solidarity, understanding, and cooperation between religions and castes.
In one of his most famous poems, Tagore argued for independent judgment and freedom of thought. His message had always been the tearing down of "narrow domestic walls" that threatened peace, stability, and progress.

Where the mind is without fear and the head is held high Where knowledge is free Where the world has not been broken up into fragments By narrow domestic walls Where words come out from the depth of truth Where tireless striving stretches its arms towards perfection Where the clear stream of reason has not lost its way Into the dreary desert sand of dead habit (from Where the Mind Is Without Fear).

So, in "The Postmaster," the main character remains unnamed. It is entirely possible that Tagore meant the postmaster to represent a universal character, one who exists within the higher reaches of the caste system and who zealously protects the status quo at all costs. In the story, the postmaster abandons his faithful servant girl, despite her expressed desire to go with him:

When the postmaster had finished his supper, the girl suddenly asked him: "Dada, will you take me to your home?"
The postmaster laughed. "What an idea!" said he; but he did not think it necessary to explain to the girl wherein lay the absurdity.
That whole night, in her waking and in her dreams, the postmaster's laughing reply haunted her—"What an idea!"

The postmaster thinks Ratan's question "absurd." While Ratan sees her employer as a father figure, the postmaster feels no answering emotional attachment to her. The question of caste (and reputation) strongly permeates this story; thus, Tagore does not name his protagonist because he is highlighting a universal issue in Indian culture, one that continues to resonate even to this day.

5) in the story Harrison Bergeson, how does the tone of the story change once Harrison appears on the television?

The tone for much of the story before Harrison's appearance is somber, sardonic, and even fatalistic. Equality is portrayed as an established reality, the tone so insistent that we are immediately suspicious of its sanctimonious certitude.

The year was 2081, and everybody was finally equal. They weren't only equal before God and the law. They were equal every which way. Nobody was smarter than anybody else. Nobody was better looking than anybody else. Nobody was stronger or quicker than anybody else. All this equality was due to the 211th, 212th, and 213th Amendments to the Constitution, and to the unceasing vigilance of agents of the United States Handicapper General.

Yet, when Harrison appears on the screen, the tone becomes animated, anticipatory, and even defiant. The earlier passive tone becomes overshadowed, and the new tone produces a brighter, more hopeful mood.

Harrison tore the straps of his handicap harness like wet tissue paper, tore straps guaranteed to support five thousand pounds.
Harrison's scrap-iron handicaps crashed to the floor.
Harrison thrust his thumbs under the bar of the padlock that secured his head harness. The bar snapped like celery. Harrison smashed his headphones and spectacles against the wall.
He flung away his rubber-ball nose, revealed a man that would have awed Thor, the god of thunder.

The once helpless Harrison becomes a sensational figure of dissent. His rebellion is executed at a frenzied pace. He and his partner leap into the air, simultaneously abandoning the "laws of the land" and the laws of gravity and motion. The tone, although changed, invites our skeptical reaction. We are led to question how long this euphoric state can last.
The dancers leap almost thirty feet high, skimming the studio ceiling as they do so. Then, hovering in midair, both lean in for a prolonged, passionate kiss. They are brought back to earth when Diana Moon Glampers fires twice from her "double-barreled ten-gauge shotgun." The tone then reverts back to the resigned, somber, and fatalistic one we see at the beginning of the story.

Can anyone from a lower social class ever feel content despite one's humble beginnings or must someone have money to feel content in life?

Your question asks whether someone who comes from a humble background and lives within the parameters of a lower social class could live satisfied or content regardless of this limitation. It also asks whether contentment in life is achieved by having money.
Here is an excerpt from an article by Robert Kenny, published in the American Psychological Association Journal: 

Extremely wealthy people have their own set of concerns: anxiety about their children, uncertainty over their relationships, and fears of isolation

What is entailed in this comment is that even people with large supplies of money can limit themselves, within their own worlds, to live unhappy lives full of worries and anxiety over things that they cannot really control. Therefore, no matter how rich they are, if they do not know how to use their resources (both monetary and non-monetary) wisely, they will end up living a life of poor quality, amidst the riches that surround them. Ultimately, it is the individual—not his funds—that determines the quality of his or her life. 
People who need money, however, may disagree with this. They may find it ridiculous to think that money cannot buy happiness. It can certainly pay for utilities, mortgages, college degrees, good transportation, traveling, and better food. However, all of the above are needs that can be met with less extravagant choices and making better money decisions. In this case, money cannot buy happiness itself, but administering what we do have with cleverness can help us get what we want without complications. 
The argument could be that it all depends, not on what you have, but on how much you actually need. There are people that can live happy lives without ever learning the difference between wealth or poverty, simply because they have learned how to satisfy their needs and be happy with that.
Notice that some individuals do not care about the brand of car that they drive, what company they work for, their job titles, or even the size of the place where they live. They simply need a roof over their heads, a method of transportation, food on the table, and a source of income to afford it all. All of this is possible without the need to have a lot of money. This makes a lot of people, as your question states, "content" and satisfied. Money, in this case, is one of many needs. Needs and wants are different things.
Therefore, people who live overly concerned about superficial things may be quite unhappy no matter how much money they have. The reason is that they have never needed anything and, as a result, they end up merely wanting things. They use their money and resources to fulfill a momentary caprice and, as such, they will never find anything to fully satisfy them. Once the caprice is gone, they are back to square one, wondering what to want next. This is far from happiness. This is a perpetual cycle of dissatisfaction. 
In "Boule the Suif," the people in the traveling party that are wealthy are only rich in terms of material things. In contrast, they are quite poor when it comes to morals, respect for others, empathy, and even fairness. All of the characters are opportunists; they all take advantage of the kind and generous nature of Elizabeth Rousset (Boule de Suif). They eat her food, pretend to like her, and then turn their backs on her after demanding that she gives herself to the Prussian officer so that they can get to Dieppe. They are quite selfish.
She, on the other hand, a prostitute deemed to be of a "low class," is very rich in spiritual gifts: she is true to herself, kind, honest, and giving. She even abides by the demands of the passengers and ends up engaging intimately with the officer, as they tell her to do. All of this was for nothing; they betray her and treat her very badly, leaving her in total despair after she had given them so much

And Boule de Suif still wept, and sometimes a sob she could not restrain was heard in the darkness between two verses of the song

In all, happiness is a matter of wants versus needs. Keeping track of our needs helps us remain satisfied and not a lot of money is needed to meet basic needs. It is a mindset that we initiate. If we learn to meet our needs and be clever about how we use our resources, we will always feel like we have enough or even more than we actually do. 
On the other hand, if we live our lives like the shallow characters of "Boule de Suif," we will find ourselves leading pretty empty lives with a very poor moral compass. Those are things that money definitely cannot make better in any way. 
http://www.eastoftheweb.com/short-stories/UBooks/BoulSuif.shtml

Calculus: Early Transcendentals, Chapter 6, 6.2, Section 6.2, Problem 3

The volume of the solid obtained by rotating the region bounded by the curves y=sqrt(x-1), y=0, x= 5 , about x axis, can be evaluated using the washer method, such that:
V = int_a^b pi*(f^2(x) - g^2(x))dx
Since the problem provides you the endpoint x = 5, you need to find the other endpoint of interval, hence, you need to solve for x the following equation, such that:
sqrt (x-1) = 0 => x - 1 = 0 => x = 1
V = int_1^5 pi*(sqrt(x-1) - 0)^2 dx
V = pi*int_1^5 (x - 1)dx
V = pi*int_1^5 (x)dx - pi*int_1^5 dx
V = (pi*x^2/2 - pi*x)|_1^5
V = (pi*5^2/2 - pi*5 - pi*1^2/2 + pi*1)
V = (25pi)/2 - pi/2 - 4pi
V = 12pi - 4pi
V = 8pi
Hence, evaluating the volume of the solid obtained by rotating the region bounded by the curves y=sqrt(x-1), y=0, x= 5 , about x axis , using the washer method, yields V = 8pi.

Describe the effect of European exploration and colonization on African and Native American cultures. How did each group react to confrontations of societies?

A basic element of the mythology Europeans created about both the African and Native American cultures was that these people, before European contact was made, were basically "savages" and lived in a primitive, non-civilized condition. This then justified whatever kind of exploitation and mistreatment that was carried out, from 1492 on. It is a good idea to look at Thomas Paine's essay "African Slavery in America," written in 1774. Paine states,

The managers of that trade [i.e., the slave trade] themselves, and others, testify that many of these African nations inhabit fertile countries, are industrious farmers, enjoy plenty, and lived quietly, averse to war, before the Europeans debauched them with liquors, and bribing them against one another; and that these inoffensive people are brought into slavery, by stealing them, tempting kings to sell subjects, which they can have no right to do, and hiring one tribe to war against another, in order to catch prisoners.

The European slave-trade destabilized the economy of western Africa and inhibited it from benefiting from the technological advancements which were made possible in Europe at least partly by the exploitation of resources and people from non-European lands.
The situation was somewhat different with the Native Americans, but similarly, their existing civilization was dismissed by the Europeans as a form of savagery. In addition, huge numbers of the indigenous Americans were wiped out when they came in contact with Europeans because they had no natural immunity to diseases such as smallpox, which the Europeans carried. The coming of the Europeans was seen as an apocalyptic, almost supernatural event by many Native Americans. The Europeans' weapons, body armor, and horses, all of which had never been seen before on the American continent, made the Europeans appear superhuman or even godlike to some. Two books which are worth consulting for an analysis of the contact between Europeans and the indigenous people of America are American Colonies, by Alan Taylor, and The Invasion Within, by James Axtell.
An aspect of the differing reactions by these two continents to the Europeans is rooted in the fact that Europe did not at first send its own people to "colonize" Africa as it did America. Only later, in South Africa, parts of North Africa such as Algeria, and a few other areas did this happen on a more limited scale. One might consider whether this is the only reason that eventually the African countries were able to form independent nation-states as they are today, while the indigenous Americans were not. The latter were largely the victims of either unintentional (because of the diseases alluded to) genocide, or in fact deliberate genocide, with the relatively few remaining people dispersed, placed on "reservations" or "reserves," or intermarried with people of European and African descent.

Colonization in Africa devastated the continent; from the earliest days of international slave trade in the 17th century to the Belgian presence in the Congo during the 20th century, European colonization resulted in the exploitation and death of African people. The slave trade vastly decreased the population of Western Africa and led to constant war between some nations, due to the economic benefit of selling prisoners of war as slaves to European traders. Enslaved Africans were treated very poorly and often violently abused by their owners and separated from their families. In the 20th century, the "scramble for Africa" led many European countries to claim land in the continent in order to exploit its natural resources. One especially brutal example of this colonization was the Congo Free State, most of which is the Democratic Republic of the Congo today. Belgian King Leopold III used African slave labor to extract the country's ivory and rubber, and the violence of Belgian officials combined with the diseases they spread in the country lead to a death toll that may have constituted almost half of the Congo's population prior to this colonization.
Similarly, European colonization in the Americas resulted in widespread violence, spread of disease, and population decrease for the Native American people. Initially, Native American people aided and traded with the European explorers. However, as they began to settle in the Americas, Europeans began to spread diseases like smallpox which the Native Americans had no immunity to, displace them from their land, and defy treaties. As the U.S. expanded in the 19th century, Native Americans experienced continual displacement. Additionally, many served in the Civil War in the hopes that their service to the U.S. would benefit their people, but soon after, the government passed the Indian Appropriations Act that denied any autonomy to Native American nations. Also in the 19th century, Native American boarding schools were established to take Native American youth from their homes to assimilate them into white American culture.
In reaction to European colonization, Africans and Native Americans practiced some forms of resistance. In the 19th century, Africans began using armed military engagement as well as forms of guerilla warfare against European colonizers. Some African countries managed to drive away European colonizers altogether; for example, the Ethiopian army gained their land back from the Italians and successfully resisted colonization. Native Americans also fought a number of wars against the Europeans, such as the Battle of Little Bighorn and the battle at Wounded Knee. These acts of armed resistance were largely unsuccessful, as many more Native Americans than European Americans died, and they continued to experience displacement and violations of land treaties.

How is Gordimer’s inclusion of the description of the miners also an explanation of why people are afraid in South African society?

This story was published in 1989, in the waning years of the apartheid in South Africa. The apartheid was a system of racial segregation in South Africa which lasted from 1948 to 1994. During this time, a white minority controlled and oppressed the country of a black majority. In these waning years, crime and protests were common. In this story, Gordimer (a white South African) reflects fears of crime and violence which many people had under this system and as it headed towards transition.
She awakens to a mysterious sound and fears an intruder is in her house or trying to get in. This fear she illustrates alludes to the uncertain future which South Africa would have after the apartheid would be abolished. However, she comes to her senses and realizes that the noise is simply a creaky house which had been built upon an underground mine. So, the inclusion of the miners actually suggests a moment of logic, reason, and maybe even hope. Realizing her fear was misplaced, Gordimer indirectly suggests that any racial fear of "the other" is likewise misplaced. This is only one interpretation. But, in any case, her realization about the mine under her house dispels her fear of an intruder.