I can't answer your main question about the role of women, but the Mongols were not a hunter-gatherer society. They were (and largely still are) nomadic pastoralists. That is, they subsisted primarily off herds of livestock which they pastured by continuously moving from camp to camp. I am not sure why that would have anything to do with the role of women, though.

You might be interested to know that Jack Weatherford, who wrote The Secret History of the Mongol Queens, is scheduled to do an AMA in this subreddit tomorrow. Perhaps you could ask him your question.