In the end, I don't think it matters what the gender of the participants is. But if for some reason the site could better serve people of either gender, it'd be interesting to know what tactics could accomplish that.
As a man, and a not particularly competitive one, I don't know if what chafes me about it is also what chafes women, but I think there is definitely room for other interpretations of social question answering. Either deep changes or simply different reward schedules, combined with a different initial community, ought to produce a site with a completely different "feel." Whatever it is that women are reacting to about Stack Overflow, there is bound to be a different solution they would like better, and the competition between sites could be beneficial.
For example, one problem reported by the women that were asked about Stack Overflow was that one-upsmanship was an unpleasant theme. In what way could it be made less competitive without lowering the quality of the submissions?
I think this is a wrong assumption. Qualitatively, it should be obvious that a diversity of perspectives and backgrounds would yield more answers. (Also a larger group of contributors can post more content to the site.) Empirically, multiple studies have shown that diverse groups are better at creative problem solving. Example: https://www.npr.org/templates/story/story.php?storyId=685821...