First, I'll be honest and agree to the consensus that a female Genghis Khan won't change the Jin Empire in all likelihood. And what does it mean to have a "female Genghis Khan" anyways? Are we talking about Genghis Khan developing complete androgen insensitivity syndrome in utero?
However, there are a host of misconceptions here.
Won't mean much of anything really, dynastic marriages are really only a European and Indian thing. ThIs would just end up a funny footnote of history/silly romantic movie about the orphan daughter of a Mongol Chieftan falling in love with and marrying the Chinese emperor.
To answer your questions no and no. The reason the Mongols expanded so fast was largely because of a superior military system and units (and it's going to be a cold day in hell the day the Chinese adopt the ideas of "barbarians"). The state won't expand Because as far aa i know in China women weren't allowed to be generals or lead troops. This will also probably hurt whatever emperor she marries because of that very same xenophobia.
That's only partially true. Dynastic marriages didn't happen often in Chinese history, but they did occasionally. In the Han Dynasty, the practice of
heqin meant dynastic marriages in the other direction, with Chinese women married to barbarian Xiongnu leaders. Afterwards, examples of this decline later. For example, I could point to
Empress Ashina, the daughter of a Turkish khan, err, khagan, who was married to Emperor Wu of China's Northern Zhou Dynasty. However, said Emperor was not ethnically Chinese, so this can be debated whether he was a Chinese Emperor or not. The Ming Emperor
Yongle was possibly the son of a Mongol consort, though debate continues. However, in the right circumstances, it's not impossible for this female Genghis to make her way into the Jin Emperor's harem, then maybe she could become chief consort, which is unlikely given her presumably low-standing, unless in this timeline Yesugei becomes a major Mongol warlord in need of this marriage alliance. From there, she probably does nothing, thanks to gender roles of the time.
The Jin Empire was founded by Jurchen horse nomads. It's not impossible for them to adopt Mongol horse tactics themselves, but they certainly won't do so because the Empress Consort suggests it.
Some women did lead armies or troops in China. I can only think of Princess Fuhao, Mother Lü, and Princess Pingyang as Chinese examples.
For nomadic tribal examples, I could point to the
mother of Emperor Shengzong of Liao, who had her own army of 10,000 cavalrymen, as well as another Liao Empress who. There was also the Empress Yingtian, who had her own
army, though I'm not sure she led it or not. However, neither were ethnically Chinese in today's sense, but then again, neither were the Jin Emperors.
Were the Jurchen Emperors of Jin also Xenophobic?
I see what you mean, but xenophobic is too strong of a word. However, by this time, the Jin Emperors were almost exclusively born to Chinese women, so this can be taken as a sign that the Jin Dynasty was becoming sinicized and adopting this supposed Chinese xenophobia. That's one of the reasons I don't see this female Genghis playing a major part in the Jin Empire. The Chinese women already had the market cornered on Jin consorts.
Yes, virtually every Chinese emperor was highly xenophobic. It was an ingrained part of Chinese culture that foreign influence on China was a terrible thing which would corrupt it in intrinsic ways, the Han where probably the least xenophobic and even then it's doubtful they would go so far as to adopt military traditions from barbarous nomads.
That's an exaggeration. Also, this goes back what you define as a "Chinese" Emperor. The Han Emperors learned to adapt to the Xiongnu by forming their own cavalry armies, though I'm not sure how close this was to the Xiongnu style of warfare. The Tang Emperors, considered Chinese by just about everybody even though they were initially Xianbei in ancestry, were very welcoming of foreign culture and trade. The Song and Ming Emperors never tried to get rid of Buddhism, foreign religion as it was, and participated in foreign trade to varying degrees at time.
There were also the Yuan and Qing Emperors, but they're not always considered Chinese Emperors, and I won't argue the point here.
Leaving even that aside this is a woman trying to take over parts of ruling the state from a man, which Confucius clearly says is bad and shouldn't be allowed.
That was the ideal. Historically, some women did rule China (well, only one ruled in her own name), but the Jin Dynasty is unlikely to be one where women wield power behind the scenes. So, that's why this scenario about a female Genghis goes from being unlikely to implausible.