The Tang Code (唐律), codified and used during the Tang dynasty (618-907), laid out a number of punishments for women who committed sexual acts outside of marriage. While the Tang Code itself was largely replaced by the legal codes of succeeding dynasties from the 11th century onwards, the Tang dynasty definitions of extramarital sex as criminal offenses persisted until the early 20th century.^(1) To say that sex outside of official marriages was heavily stigmatized would be a slight understatement and it wouldn't be until the last ditch legal and governmental reforms undertaken by the Qing dynasty in its final years that such legal definitions were dispensed with. It should also follow then, that to bear a kid out of wedlock was an immense shame, even for a woman of high social standing. It would be very difficult to support a child without a male figure and unless a woman came from a well-off background and was not castigated by her clan, survival with a child would have been an immense challenge. I do not have the sources or figures to address the following question, but it would be interesting to read a study or analysis on the demographics and familial statuses of orphans in imperial China.

Divorces are complicated. Divorces were by and large initiated by the husband and children were rarely allowed to follow mothers out of a family if such an action was allowed to proceed after a period of civil investigation. I can't really answer questions 3 and 4 for you, but I can tell you that it would have always been more bearable for those with means than for those without. Among the rural poor in the late Qing, even widows, a group that could and did receive social support at the time, struggled to survive during times of hardship.^(2) Mothers who were unmarried and had weaker social connections upon which to depend probably suffered at least as much. I do regret that I don't have any sources and especially any primary sources to give you a more satisfying answer regarding mothers who bore out of wedlock or mothers who went through divorces though.

Widows were in a better albeit similarly precarious situation during the time period. Widows who had families to look after actually experienced a bit of a decline in their fortunes in the late imperial era. During the Southern Song dynasty, the issue of remarriage was less controversial than during the late imperial dynasties. By the time the Ming and the Qing came about, attitudes had been shaped by harsh orthodox moral commentary and debates.^(3) The education women would receive in the late empire (formal if they were well-to-do, but by-and-large informal for the vast majority of women within the empire) covered the virtues of righteous widowhood for a start. Lan Dingyuan, a Qing official, writes in 1712 that "the basis of government of the empire lies in the habits of the people, and the surety that their usages will be correct is in the orderly management of families, which last depends chiefly upon the women... Then teach her, in times of trouble and in days of ease, how to maintain her purity, how to give importance to fight principles, how to observe widowhood..."^(4). His writing is but a small part of the late imperial historical record that supports this view of 'righteous widowhood'. Widowhood does at least have some precedence within the Chinese classics (Mother Meng within Mencius continues to be upheld in China as an upstanding model of motherhood and widowhood) and under the correct circumstances, it was not unfeasible for a widowed mother to continue being regarded highly within her community during the late empire.The Chinese feminist Liu Jihua notes that during the Yuan, Ming, and Qing dynasties, legal and social sanctions began to take form within Chinese society and for the first time in Chinese history, the Ming court published a series of regulations to either punish or reward widows based upon their merit. Such merit or demerit largely revolved around how well they were able to rear their children and whether or not they remained chaste after the deaths of their husbands.^(5) So as you could imagine, if a woman was particularly chaste and raised wonderful sons, she might even be praised by the emperor which did actually happen on occasion. On the flipside, the social expectations set for widows (chastity and devotion to her passed husband's children and family) could lead to ostracism from certain social elements should a woman choose to remarry or otherwise stray from the straight and narrow. The semi-imagined but broadly biographical story of Lan Sanxiu comes to mind. Lan was a highborn woman from a scholarly family who was widowed over the course of the Ming-Qing dynastic transition in the 1640s. After the turmoil, she is eventually wedded to a Manchu prince for whom she bears a son. This results in great favor from the new Qing ruling family but does lead to some ostracism from her own family as she broke her social obligation as a widow and in a way turns her back on her original marriage.^(6) Obviously, Lan still does extremely well as she becomes the principal consort to a princely household but the social stigma associated with 'improper' widowhood transcends class boundaries.

To sum this all up, I think the ultimate answer I can give you is the extremely unsatisfying "it depends". The passage from the Cambridge Histories I draw from notes that widows, orphans, and the childless were starving to death in the late 19th century. Regardless of how they and their children were regarded within a community, poverty is poverty. The story of Lan shows that families may become resentful if an individual strays from the orthodox path, despite how powerful or influential they eventually become. It is doubtful that as a princess, Lan had to worry about starving to death within the princely household yet her brothers shunned her for marrying after the death of her first husband. Raising any kids she might have had from that first marriage may have been socially difficult, if not economically. These factors would be amplified for mothers who bore children out of wedlock and thus existed on the periphery of social acceptability.

^(1) Alison Sau-Chu Yeung. "Fornication in the Late Qing Legal Reforms: Moral Teachings and Legal Principles" in Modern China vol. 29 no. 3. 297-328

^(2) "The Agrarian System" in The Cambridge History of China vol. 13. 230-269

^(3) Katherine Carlitz. "Shrines, Governing-Class Identity, and the Cult of Widow Fidelity in Mid-Ming Jiangnan" in The Journal of Asian Studies vol. 56 no. 3. 612-640

^(4) "Lan Dingyuan on the Education of Women, 1712" in The Search for Modern China: A Documentary Collection. 74-76

^(5) Susan Mann. "Widows in the Kinship, Class, and Community Structures of Qing Dynasty China" in The Journal of Asian Studies vol. 46 no. 1. 37-56

^(6) "From Orphan to Princess: The Story of Lan Sanxiu" in Voices from the Ming-Qing Cataclysm: China in the Tigers' Jaws. 93-113