S'mon said:
One good model for a matriarchal society is the lion pride one, which also fits some real world human societies quite well. Basically the females do the work, effectively head the family groups, organise the societal structures. The males laze around and fight each other.
During the day, this is what the tourists see.
At night, a completely different lion society emerges, where males and females hunt together, and different pride members are 'in charge' depending on what they are hunting, as the different pack members specialize in different prey species and end up taking the lead in those situations. (Something rarer in the human world, where the least-qualified for a given circumstance may well be the 'leader.')
Since the males feed first on these nighttime hunts, the females aren't as 'full' during the day, and sometimes get up and kill something during the day, when lions don't traditionally hunt, and the males sit back and watch, bored, only getting up when the kill is complete to take a symbolic 'first bite' before wandering back to continue their daily nap.
The males aren't lazy, they're just selfish and end up hogging the best meat for themselves in the nightly hunts, leaving the females hankering for seconds later in the day.
As for the general topic, I don't have any gender roles set for gnomes, portray halflings as much like rural humans, and various dwarven clans fluctuate from gnome-like gender blindness to human/halfling like gender roles, which can sometimes be cause for animosity between different dwarven lines ('Your females are disrespectful!' 'Well if you treated yours better than cattle!').
Elves are insanely protective of their womenfolk, with the adventuring elven female being essentially a runaway from smothering elven society. Elven women are encouraged to, at best, study the arts of archery, so that they can stay out of physical combat. Elven men are encouraged to learn swordplay and fabulous displays of 'bladesinging' or 'sword-dancing' or other flowery displays to both impress their womenfolk and also encourage them to stand between them and the dangerous front-lines (There are practical sides to this. Elven 'Bladesingers' practice combat expertise and sit at the front lines defending for all they're worth, while their wives and sisters and mothers and daughters rain down death upon their foes from the trees. It's not *all* pig-headedness...). Elven women range from quietly tolerant of this chauvanism to quite fed up with it, which explains elven adventuring females almost entirely, 'fragile young flowers' who have grown sick and tired of the patronizing of their elven males and go adventuring to get away from them.
Kobolds have no gender disparity. The females lay eggs, which are kept warm by rotting vegetation or warm sands or something and need minimal tending, so the she-bolds are right there on the front lines fighting with the he-bolds.
Orcs keep their womenfolk back in the caves tending the young, uneducated, pretty much beaten into submission.
Hobgoblins do the same, but actually have status for their women, and a female Hobgoblin is going to consider herself the forge that produces warriors, much like the Queen of Sparta, and while the male hobgoblins run certain aspects of society, the females have their own specific areas in which they are in full control.
Gnolls are matriarchal, with the females beating the crap out of, and killing, and *eating,* any male who questions their status.
Bugbears and Goblins, despite the whole child-rearing thing, tend to barely recognize gender differences, and rule is determined and respect gained by acts of viciousness, not gender. Physical strength is roughly equal between the genders of both races anyway, and is not relevant in the staggering acts of treachery and betrayal they commit upon each other to gain advantage.