Within Thedas, there are several different views with regard to sexuality and marriage. Common to all groups is that marriage is not primarily regarded as a romantic affair, but a duty to one's family.
According to the Chant of Light, Andraste had a spiritual marriage with the Maker. All Andrastian priests are therefore symbolically wedded to the Maker and sworn to celibacy. In the eyes of the Chantry, marriage is both a celebration of tradition and a practical decision. Through a favorable marriage, one can make valuable connections, improve one's social standing, and secure financial stability for oneself and one's family. Love is not necessary for a successful marriage but is nonetheless desirable, as it makes the union stronger. The Chantry permits annulments but not outright divorce.
Among city elves, marriage is a rite of passage, the single greatest thing that distinguishes children from adults. Elven children are usually matched by their parents and the Alienage elders, and the bride and groom often will not have met each other before the marriage ceremony. Betrothals are often made between families from different Alienages in order to promote inter-city trade and relations, as well as to bring new elves into the Alienage community. Marriage ceremonies require the approval of the Chantry, and are officiated by one of the Chantry priests. In a community where there is mostly little cause for celebration, a wedding is a tremendous affair, and the whole district will turn out to enjoy the feasting and dancing.
Not much is known about the Dalish, and the same holds true for their views on sexuality and marriage. The Dalish appear to be more in line with Fereldan thinking rather than other cultures like Orlais. This meaning that they tend to take relationships slowly and seriously with each other. Once they have come of age, they may choose a mate of the opposite sex with whom to bond for life. It is unknown how they view same-sex relationships, but there seems to be no stigma against it other than for the fact it will not produce a child for the dying race. As with most cultures, the Dalish do not approve of interspecies romance, and it is forbidden by them. To even dally with a human could earn exile from one's clan. This likely stems from both past hatreds and the fact that "pure elves" or Dalish are becoming increasingly rare.
Dwarves of Orzammar
The dwarves of Orzammar are noted for their polyamory (the keeping of multiple intimate partners), especially among the upper classes. While a dwarf has only one legal spouse, many nobles keep concubines, who are considered part of the household and have their names recorded in the Memories. A dwarf's caste is determined by that of their same-sex parent, so many commoners offer themselves to nobles as sexual partners in the hopes of producing a child of the noble's gender. As dwarven fertility is in dangerous decline, any noble child is considered a blessing, and such offers are rarely turned down. If such a union produces a child of the commoner's gender, however, the infant will be disposed of, as it would be part of the lower caste and an embarrassment to the noble parent's house.
Although Grey Wardens are not forbidden from marrying, it is rare for them to do so. Their short lifespans make them poor candidates for parenthood or long-term relationships. Even if they are not killed in battle against the darkspawn, they rarely survive past middle age due to the darkspawn taint. Most Grey Wardens dedicate their brief lives to fulfilling their duty, leaving them little time for pursuits such as family life. Furthermore, the taint makes conceiving a child very difficult (nearly impossible if both partners are Grey Wardens).
Mages, because of their outsider status, are not bound to traditional social mores and consequently enjoy greater sexual freedom than most, though the exact degree seems to vary from Circle to Circle. They are discouraged from marrying and reproducing, as the offspring of mages are very likely to possess magical ability themselves. Dialogue between Wynne and Alistair hints at some form of birth control or abortion being readily available to mages, perhaps to discourage the propagation of magically gifted children. If a mage does produce a child, it is taken from the parents at birth and raised under the supervision of the Chantry. Those children who prove to have magical ability are immediately transferred to a Circle of Magi in another country, where they are held in complete isolation and taught to control their powers away from their birth parents. Mages (and non-mages) who do not wish to be separated from their children will sometimes conceal them from the Chantry and either train the child themselves (as Malcolm Hawke did) or hire an apostate as a tutor (as Arlessa Isolde hired Jowan to tutor Connor Guerrin).
Aristocrats in Orlais are notorious for their hedonism and extravagance, and sexual relations with multiple partners of either sex are not uncommon. The peasantry is much less tolerant of such behavior, at least in public. According to The Stolen Throne, the Emperor Florian is rumoured to reputedly had an affair with his cousin Meghren, which resulted in Meghren being banished from Orlais to Ferelden. The scandal surrounding their relationship was likely due to their being related, however, not to the fact that they were both men.
Qunari don't have "family units". They don't marry or choose partners. Qunari do not even know to whom they are related. A Qunari's "family" are his or her coworkers.
They also don't generally associate mating with love. They feel love. They have friends. They form emotional bonds with one another. They just don't sleep with each other to express it. And if they do, they get re-educated by the Ben-Hassrath. If such a thing occurred and produced a child, the same thing would happen to the offspring as happens to all other Qunari offspring: It would be raised by the Tamassrans, evaluated, and assigned a job. Qunari don't waste people unnecessarily.
Whether same-sex attractions are frowned upon or otherwise by the Qunari is currently unknown.
Same-sex relations are generally considered strange, though not inherently immoral, in Ferelden. Orlesians regard homosexuality as a mere quirk of character, and the Antivan Crows show a winking tolerance for (if not encouragement of) relations with multiple partners of either sex. The Chantry does not seem to have an official view on the subject. There is pressure in certain circles, such as the elves and the human nobility, to marry an opposite-gendered partner, but this is motivated by pragmatism rather than morality; a homosexual couple cannot have biological children. For a dying race like the elves, it is vital that every fertile individual produce offspring.
Interspecies relationships are possible in some circumstances, but frequently frowned upon by one or both of the parents' cultures. Elves, for example, are bound only to produce children with their own kind because their race is dying, and a child of such a relationship would be pure human. Relationships with humans are considered traitorous, and any elf who conceives a child by a human risks expulsion from his or her community.
Dwarves and humans can also reproduce (such a union will produce a half-dwarf), but this is very rare, as dwarves consider humans and elves to be lesser races.
Brothels are legal in Ferelden, Kirkwall and in the Dwarven city of Orzammar. Although they are not considered respectable institutions, it is no more taboo to visit them than a seedy tavern or gambling house.