Why there are more people of red hair among Arabs than Mongols when both populations are naturally born with dark hair?

This question seems too contrived to have any sincere intention, such as, " ... knowing how ravenous they were, wouldn't they also have wanted to take women back with them to Mongolia?" Did OP mean to assert they were more ravenous for women? Than whom? Actually, no, I don't know how ravenous.

All conflicts - from ancient to modern era - results in suffering of women, children, and the generally weaker groups (tribes). So, as Mark Wallace suggested, the phrasing is, in my opinion, reckless.

If OP is seeking a reasonable question I could probably answer it straight forwardly. As it stands, and as Mark suggested, it would be better to clean it up because it is not answerable.

So, presuming OP truly is doing some research, the answer to one of the questions by OP, "whether the women gained influence in the courts", I take it the question meant whether women slaves gained influence in the courts. The short answer is "No". And it isn't because they were women, but because they were non-Chinggisid.

My hope is OP puts more care into drafting the question, and aim for clarity. I'm quite certain that many here can provide can very reasonable answers once that's done. I, too, will attempt with a more detailed answer if this is done.

Contingent Answer

This is contingent on the fact that OP wanted to ask this question, and my revision of the question is correct.

From above: I take it the question meant whether women slaves gained influence in the courts. The short answer is "No". And it isn't because they were women, but because they were non-Chinggisid.

First, womenfolk who were enslaved were still treated with some measure of respect (as part of the appanage). In other words, they were property of the Great Khan (Chinggis Khan) and cannot be disposed off improperly.

Second, slave women were eligible for marriage. To avoid incest and re-productive issues, the Mongols practised exogamy and the enslaved womenfolk played a role in this.

Finally, to be influential at the Mongol court, one normally had to be a Chinggisid and women had real influence, especially during power vacuums, between selection of Khans, whilst waiting for the kurultai's final decision. See Chapter 2 (Regents and Empresses: Women’s Rule in the Mongols’ World Empire) in "Women in Mongol Iran" (Edinburgh University Press, 2017). Open access book.

(Because LangLangC requested)

Contingent Quotes

Let's wait for OP to clarify the question.

Meanwhile, here's two short quotes. Both from The Cambridge History of Inner Asia - The Chinggisd Age (Cambridge UP, 2009), pages 28 and 94, respectively:

This was a world charcterized by endemic feuding and raids in which the livestock of neighbouring clans or tribes were looted and individuals or even whole clans were carried off as prisoners. Those enslaved included both womenfolk, who were a means of perpetuating well-established exogamous traditions, and males who were employed by their new masters in a menial capacity. Such 'long-standing serfs' might in time rise to positions of responsibility, though less often than free warriors who moved between tribes in order to attach themselves to a successful war-leader as his sworn retainers (nokod).

NOTE: The nokod (singular, nokor), or "companions," of major lineage chiefs or tribal khans were important estate in medieval Mongolian society. More info role and signficance of nokod here, "The Cambridge History of China, Volume 6: Alien Regimes and Border States, 907-1368" (1994).

Both the conquered and the military rank-and-file were the property or vassals of the Chinggisid dynasty and the Mongol aristocracy to whom they were allocated, but allocation was temporary and there were reallocations of troops from one appanage to another and permanent deportations of conquered subjects to the dominions of the paramount leader.

In other words, as a tribal society, all conquered subjects were property (which requires further explanation for greater understanding in another place) and therefore, and important resource to Mongol society.

(This 'contingent' approach is clearly silly but, hopefully, it illustrated the need for clarity in questions).