Many aspects of societies in equilibrium can be derived from two things. One is the question: what prevents further increase of the population? The other is the shape of the food production isoquant curve.
These occur when population is kept down by diseases, not famine. As a general rule, this means that food is cheap as comparatively little labor is sufficient to produce enough food for everyone. After all, land is relatively plentiful—there is no need to substitute labor. In extreme cases, land (and food production capacity) is a free good like air.
The most important consequence is that single mothers can feed themselves and their children. Since paternal resource input is of little effect, men don't do it. In general, man-woman relationships are shallow and often not long-lasting. (While this makes the concept of age at "marriage" ill-defined, in general, women start reproducing early.) In turn, men aren't neurotic about spending effort on someone else's child, thus women have sexual freedom.
As such, men—not having to work hard for sustenance—tend to go off into a "frat house" and spend their effort on various forms of competition for status (and access to women). "Sports", language one-upmanship (e.g. rap), art, violence, raiding neighboring tribes, gossip.
If diseases can't kill people fast enough, population density goes up. The quantity of food to sustain each family must come from less and less land. Thus society must slide along the food production isoquant and use more and more labor to produce each unit of food.
In these societies, the curve theoretically continues sloping down beyond the quantity of labor that can be put forth by each family. As society grinds against the limit, single motherhood is a death sentence, more or less. Monogamous marriages are durable, as paternal input—the food grown by male labor—is required essentially until the children reach adulthood.
This makes men quite neurotic about paternity. This does not lead to particular oppression of women, though, as the marriage market is evenly matched, since the limit of food production (and hence reproduction) is labor. Women's capacity to work is economically valuable in itself, thus their parents are more than happy to let them stay and contribute. Comparatively late marriages and a custom of the groom buying (or notionally stealing) the bride are to be expected.
These societies are a bit drab. For one thing, the vast majority of available labor is sunk into food production, and there isn't too much left over. (Imperial taxes can divert some labor to other purposes.) For another, there is little point to fancy displays, as polygamy is reproductively pointless. To be very blunt, one woman can give birth to as many—indeed more—children than a man can feed. Hence the balance in the market: any man who can work can expect to find a wife, and any woman can find a husband.
"Imperial taxes". What imperial taxes?
If the society in question lives on relatively flat and easy-to-traverse ground, at high population density, has developed expensive but very effective weaponry—and cultivates cereal crops—then a warrior ruling class, whether internal or external in origin, can extract a sizable fraction of the labor.
They can use this labor to "live like kings", erect monuments, temples, palaces. Some would include cities—cities, which can be feasibly supplied with food—on the list, and while they are indeed characteristic of these societies, they develop on their own for economic reasons, rather than as a result of coercive extraction.
This ruling class can also be the exception and engage in polygyny, often to an absurd degree.
On the other hand, if the terrain is hard to traverse (mountainous or marshy), population density too low (despite butting against the Malthusian limit), a revolt difficult to defeat in battle, or the surplus hard to extract, the population might not be subjugated.
Typically, this means the population will stay poor. In the absence of cities (since transporting food is too difficult), trade, and the division of labor these allow, productivity in material objects is low. And in any case, most labor goes into producing food anyway.
However, this freedom also means that a very large fraction of the population carries weapons. This causes no concern to whatever political leadership exists, since the political organization is usually based on clans. (Some degree of polygyny is possible.)
This small size of polities leads to ubiquitous internal warfare—two clans raiding a third for movable wealth and women, occasionally slaves (labor is valuable, raising children from scratch is expensive). At the border to a richer, lowland empire, with a subjugated population, often this warfare spills over, and highland clans raid the disarmed imperial subjects. (One imperial solution is to set up marches, a zone where semi-free subjects don't pay taxes, instead they have an obligation to carry arms and defend the empire from barbarian incursions.)
In some cases, the food production isoquant abruptly ends before reaching the limit of labor. (Or it doesn't end per se, just the marginal product of more labor is nil; the curve is horizontal.) "You can't herd goats harder." Here, population is limited by the availability of food, without any way of producing more.
Land, livestock and other variations on Magic Box of Food become, with little exaggeration, the most valuable things, and their ownership determines everything. This crucial asset is usually owned by (groups of) men, since they are the ones who can fight to defend it (and could fight to acquire it).
This leads to an imbalance in the marriage market. (Again, monogamy is the rule, as food, not raw reproduction is the limit.) Men who have a larger Magic Box of Food are unlimitedly better grooms than the other hopefuls. Women (and their families) frantically compete for the best husband they can get.
One aspect of this competition is that women pay large dowries for the right to marry an eligible groom. Another is that women are thoroughly oppressed, to satisfy the husband's nervousness about paternity. This is where women are completely covered in clothing, are largely confined to their parents' or husband's house, and always guarded by a male relative if they venture outside.
More sinisterly, this is where women have their genitals mutilated, and where insufficiently chaste women are killed by their relatives to retain the family honor—the credibility of the promise that the other marriageable daughters of the family are chaste. Marriages happen at a young age, since the parents have no use for their daughters in their own household. (Food is expensive, labor is cheap.)
Something of the reverse plays out at the bottom of the social ladder. There is a surplus of men with a small Magic Box of Food, and they compete fiercely to get a wife. Alas, they struggle in vain—families with daughters would damage their prospects if they married one to a lowly man. If the society is divided, it may happen that the subculture of the poor embraces polyandry.
These societies can be extremely colorful. There is a surplus of labor to be sunk into works of art, and motivations to do so—such as the brides' competition for the best grooms.
Usually an outgrowth of subjugated labor-limited Malthusian societies, but often governed by different considerations than the society they were embedded in.
Throughout history, it was extremely common for cities to far overshoot the population density where disease kills people faster than they can reproduce. This did not make them non-Malthusian, since food was still expensive, and all consequences of that apply. The most useful part of the phrase to deny is society, as the city is not reproductively self-sufficient, instead relying on a constant influx of new residents from the hinterland.
The key considerations "inherited" from the rest of society is that food and labor are both expensive. Customs motivated by these factors—e.g. strict monogamy and late marriage—survive unchanged.
The primary point of divergence is that high-productivity cities could in many cases defend themselves from the ruling class in a way that peasants couldn't. They would become effectively self-governing free domains surrounded by their hinterland ruled by the warrior class. (Formally, they are part of the Kingdom, yet e.g. the King has to ask the Mayor for the right to enter.)
Eventually, cities would often manage to depose this ruling class and come to directly and formally rule over their hinterlands. In this case, the city-state is usually called some form of republic. Even when it isn't, that is the result of urban politics; the peasants in the hinterland aren't political actors.
This "free" urban population means that when necessary, a large fraction of the population can be armed. (This frees the cities from the king's rule in the first place.) Unlike the poor mountain clans, however, citizens are rich, thus the characteristic army is heavy infantry—hoplites and the Roman legions.
"Rule of the sea". Ships are in all eras the cheapest form of long-distance, er, shipping. This fact more or less guarantees that port cities will be Malthusian urban societies.
The particularly interesting trait thalassocracies have is that, unlike nearly all previous types of societies, they have a large geographic sphere of influence. Thus they come into contact with many foreign societies, who laugh in their face when they explain their beliefs and customs. Generally, they cannot sweep this under the rug by being hostile—they need the trade relationship, unlike the more self-contained societies.
This sets in motion a process of justifying the traditions with reasons that the foreigners cannot just sweep away, because they are universal. This process will instead, eventually, create science and the industrial revolution.
Thalassocracies tend to be especially free, since many of their citizens can just up and leave.
If things go well, science and trade can break out of the Malthusian limits. Improved crops, better cultivation techniques, completely new crops from elsewhere on the planet, and settlement of previously non-civilized breadbaskets can increase carrying capacity faster than the population grows. This happens despite science lifting the population-limiting disease burden from cities, where people move as labor-productivity increases in agriculture.
Production is organized with extreme division of labor, in highly specialized and controlled environments. This pattern can be named "the sundering of the home". While in other societies, a shoemaker may sell shoes from the front of his house, make them in the back, and reside above the shop, in a modern society these functions are separated to different buildings. Healthcare, childcare, entertainment, and many other functions also move into their dedicated locations. If this pattern is carried to completion, the "home" will only serve as a dormitory.
While the analysis of the other forms of society could assume equilibrium, in this case that assumption is inapplicable. Although in principle it can happen that reproduction stays low indefinitely, the best way to approach the situation is to admit the temporary nature of the "whalefall" plenty.
Taking the first fundamental factor—food is cheap, sometimes astonishingly so—one might try to apply the strategy from the equilibrium non-malthusian society. Single motherhood is, indeed, not completely unviable. However, in this society the vast majority of people are not directly engaged in food production. Instead, they work in some precise, specialized, highly controlled environment that is incompatible with simultaneous childcare (unlike light agricultural work). Consequently, this strategy is mostly followed by the underclass who are unable to function in the high-precision environments anyway.
Strict monogamy (typically with delayed marriage) is inherited from the usually labor-limited malthusian past of these societies. (Remember, the urban and thalassocratic patterns are embedded in hinterlands of this form). While this strategy is indeed workable (coming from an expensive-labor environment), it tends to founder on excessive delaying due to productivity accumulating, which did not use to happen.
One solution is to embace the sundering of the home. Put childcare completely in its own environment, to minimize its interference with the other functions of life. (Actually complete separation would be "orphanages for everyone!"; in practice the parents still do some childcare during the night.)
Another group of solutions is to bring functions back withing a larger household. This can be multigenerational, or in a radical departure from the society's own past, based on groups of similar-aged women founding a household, with a few members caring for the children of all members (or might as well say business partners). Despite some attractive features—multiple incomes and unemployed members inherently limit the potential damage from some partners losing their jobs—multiple-income households are necessarily exposed to large "shear" forces between the members' job movements.
(The location of the household needs to be satisfactory for all employed members, even as the set of the locations of employment changes. Either the members sacrifice mobility of employment, or the household faces repeated crises of needing to relocate—if any sufficient solution exists at all! If members are unwilling to sacrifice mobility of employment, some members will probably leave the household.)
As determined by two factors: difficulty of promptly replacing the employee, and reliably observable relative differences in productivity.