You know gents, far before the Mongols even happen, here's a lot of things to consider regarding if the Mongols ever meet the Khazars (POD is during Svyatoslav's lifetime so 960s) and have their combined descendants deal with a resurgent Russia:
1. Does the Liao empire fall at all, or start to decline? Is Muzong murdered on time, earlier, never? Does Jingzong take over on time? How do the Song react? Does it focus north, or south? Does it fight with Korea? How does that fight go?
Liao doesn't fall = no Blue Mongol rise. Liao falls earlier = another tribe rises earlier to replace them. It's fraught with possibilities.
2. There is FIVE generations before the Jin become an Empire. Do the Jurchens manage to produce a powerful state in the period? If so, who is it focused on?
Is there a Jin-Song Alliance of the Sea?
3. In Central Asia at the time, do the Oghuz have good relations with Rus? Bad? Are they seeking to expand West or East?
Do they stay united or do they splinter?
Do the Kypchaks ever organise enough to challenge the Oghuz?
Do they adopt Islam (not guaranteed yet in the 10th c.)?
If they do not (cooperate with Rus to beat up Bulgaria and Khazaria, fall to the Kypchaks, splinter, then adopt Islam) there is no Turcic states in Persia, Mesopotamia or Rum as we know them, therefore it affects everything: Crusades, Rus-Byzantine trade, Rus-Byzantine alliances, Fatimids, Shia-Sunni power balance...everything.
5. Ghanzavids. What happens there?
6. Karakhanids. What happens there?
7. Russian princely wars, what happens there? For all we know, Kiev might get a Polish prince on the throne. Or the other way around.
8. Does the Roman-Byzantine religious schism even happen based on Byzantine involvement in certain politics? Do the Normans even appear on the scene as planned? Is there a papal/imperial conflict? based on that, when do the Norse adopt Christianity? Where are the bishops coming from? Do they centralise? Do they send out mercenaries like before?
....
I mean, come on. This is not chaos theory, this is not "butterflies", this is cause and effect and minimal knowledge of the situation.