How would you describe the characters of Dame Van Winkle and Rip Van Winkle, and do they reflect gender stereotypes in any way?

Dame Winkle is presented as a "termagant" and "virago," terms of derision used for a scolding, angry, verbally abusive wife. We are told that

Morning, noon, and night, her tongue was incessantly going, and every thing he said or did was sure to produce a torrent of household eloquence.

This characterization plays into the unfortunate gender stereotype of the shrew that allow Rip to see himself as a victim, and blame his problems, at least in part, on his "termagant" wife. After all, who could stay home and do work with such an ungentle wife harassing him endlessly? This stereotyping might blind us to the fact that Dame Winkle is perhaps a desperate woman, driven to distraction by a lazy husband who won't take responsibility for providing for his large family. Perhaps, if we were to see the story from her point of view, we might perceive her scolding as her way of trying to get through to him and communicate that she can't raise the children and run the farm all by herself.
However, because of her scolding, Rip is able to slot himself into the category of another gender stereotype, that of the "hen-pecked" husband. This justifies him in escaping from the house as often as possible and exonerates him from feeling guilt over his neglectful and lazy ways.
The story thus presents a dysfunctional picture of domesticity. The farm is falling into wrack and ruin, the children hardly have clothes to wear, the wife is in a perpetual state of rage, and the husband is always out wandering around with his gun or sitting at the local inn, doing nothing but talking.
Toward the end of the story, we get a poignant glimpse of Rip's surprise at seeing his home in disarray after he has been asleep for twenty years: his wife had always kept the house neat. Here, we get a brief glimpse of a woman who was, when alive, a capable housewife, but yoked to an incompetent man. Rip is lucky that Dame Winkle taught their daughter well, for she is able to care for him in her well-ordered household.