What are the roles of social class and money in Sense and Sensibility?

In order to understand Sense and Sensibility (indeed, any of Austen's works), it is critical to understand Georgian society and culture (1714-1830, includes Georgian Regency 1811-1820), especially relationships within and between classes, and to understand the laws governing the transfer of wealth, especially through marriage.
The Austen Family
It is useful to look at Jane Austen's own family’s class. Jane Austen (1775-1817) and her family were in the second class (the middle of the three upper classes, according to the 1814 "Map of English society"). Reverend Austen was an Oxford proctor, an official responsible for discipline and examinations at the university.
Rev. Austen was a gentleman, part of the gentry class, even though he owned no property. After marriage he accepted the rectorship of a large parish and opened his home to tutoring private students, like Edwards Ferrars' tutor did. His rectorship bestowed lands upon him with a "life interest" only, so he did not own the rectory land. Rev. Austen's cousin, a landed country gentleman, like Mr. Knightley or Mr. Darcy, bestowed the rectorship and rectory lands upon Rev. Austen in the same way that Colonel Brandon bestowed the clergy's "living" (income from clerical duties performed) upon Edward Ferrars.
Mrs. Austen met Rev. Austen, Proctor of Oxford's St. John's College, while visiting her uncle, Master of Oxford's Balliol College. She was related to a Duke and a Lord. Thus Jane grew up in social circles that exposed her, in the same way that Elizabeth Bennet of Pride and Prejudice was exposed, to society and culture in the two highest classes while allowing her knowledge of the third and fourth classes.
First Class / Highest Orders
Royalty, hereditary lords, like archbishops or Dukes, highest officers of the Crown, titled nobility above the level of baronet.Second Class / Upper Class
The class of landed country gentlemen, prominent clergymen (like Rev. Austen), any with great fortunes and annual incomes, and lesser title holders (i.e., baronet, knight), who represent the highest level of commoners.Third Class / Upper Class
Clergymen, prominent doctors, judges, bankers of wealth, manufacturers and merchants with large enterprises.Fourth Class / Lower Class
Lesser clergymen, professionals (i.e., lesser doctors, lawyers, teachers), lesser merchants and manufacturers, builders, shopkeepers, artists, mechanics, any with lesser incomes.Class rankings are as described in the "Map of English Society in 1814" from Colquhoun's A Treatise on the Wealth, Power, and Resources of the British Empire, 1814, available on JaneAustensWorld.com under "Social Classes in England."
Since Jane was in the second class, she, like Elizabeth Bennet, would have been eligible to marry someone like Mr. Darcy, Colonel Brandon or Willoughby. Lucy and Anne Steele are below the upper classes. They are in the fourth class, or lower class, as is made evident by their incorrect language and manners. Although we don't know about their home life, their uncle, Mr. Pratt, is a tutor who takes in private students, such as Edward Ferrars.
Jane Austen makes her most significant heroines part of the second class, although she has one heroine who is the daughter of a clergyman, and one who is the daughter of a Naval captain, both of whom were in the fourth class.
Class, Lucy, and Edward
In Sense and Sensibility, though Elinor is socially eligible to marry Edward Ferrars and Marianne to marry Willoughby, Lucy is not socially or culturally eligible to marry Edward, nor is she eligible to marry Robert Ferrars. The difference between the Steele's lower class and the Ferrars' second level upper class is too great; in strict Georgian society eligibility could not exist. This is one reason Elinor is so reluctant to believe the story when Lucy first forces her confidence upon her.
The youthful infatuation of nineteen would naturally blind [Edward] to everything but [Lucy's] beauty and good nature; but the four succeeding years—years, which if rationally spent, give such improvement to the understanding, must have opened his eyes to her defects of education, while the same period of time, spent on her side in inferior society and more frivolous pursuits, had perhaps robbed her of that simplicity which might once have given an interesting character to her beauty.
Money, Women, and Marriage
Wealth, be it property or money, is transferred in marriage in two ways: dowry and settlement. The settlement is really part of the dowry but is protected by law as set forth in the marriage contract. Marriage contracts specified the material particulars of the financial side of marriage.
Contracts included the amount the woman would bring to the husband's family's fortune and even included stipulations for such things as numbers of servants and locations of one or more houses. For example, a woman and her family might add to the marriage contract a stipulation for a house in London to accompany the man's country estate.
Dowry
The dowry is the amount of wealth in terms of cash and property that the woman brings to the man's family fortune by marriage. The financial goal of marriage in the upper classes was to expand or revitalize the man's family fortune, to provide adequately or amply for the woman's personal needs and to provide a larger inheritance for all the children of the marriage.
The husband's fortune would most often be passed on to the eldest son so the landed properties would stay intact and not be broken up to various small holdings, which would radically reduce the power of the family.
The wife's settlement could be divided amongst the several children so none found themselves dependent upon the eldest son's generosity to survive.
Settlement
The settlement was a portion of the overall dowry, not an additional amount. It was protected by law and set apart for the purpose of the woman's independent annual income from which to make personal purchases and for the purpose of keeping her from poverty after her husband's death when all his wealth would most likely be bequeathed to the eldest son. While the settlement interest was intended for living expenses, the capital was available to be given as an inheritance to the woman's children, especially to younger sons and any daughters.
A good example, and one often misunderstood, occurs in Sense and Sensibility. Mr. John Dashwood, Elinor and Marianne's half-brother through their father's first marriage, has a large fortune from his mother's marriage settlement bestowed upon him at the time of her death. Mrs. John Dashwood, Fanny, has a large settlement contractually set aside for her from her large dowry. While Fanny has independent access to the money settled on her in the marriage contract, it cannot be said that she is independently wealthy because all her wealth is owned by John Dashwood, and it is probable that her ample settlement still would not be enough to support her accustomed lifestyle.
Upon marriage, everything the woman brings to the marriage becomes part of the husband's family fortune: He gains full ownership over all the dowry including the part settled on the wife. Consequently, although Fanny has adequate means settled on her, she is not independently wealthy. Still, she is quite comfortable in meeting her personal needs and has a large capital to pass on to her son. This is the same way that John received his initial fortune: from the dowry amount settled on his mother by the terms of her marriage contract.
The son [John Dashwood], a steady respectable young man, was amply provided for by the fortune of his mother, which had been large, and half of which devolved on him on his coming of age. By his own marriage [to Fanny Ferrars], likewise, which happened soon afterwards, he added to his wealth.