Well, if I recall correctly, Temujin's (GK's) children were still pretty young at the time. Jochi was the eldest, but Chagatai was held in higher favour, so it's not unreasonable to see those too young kids being used as figureheads in a civil by the still tenuously-united clans (i.e. think Jochi's Clans vs. Chagatai's Clans). In time, expect the conflict to degenerate further until its basically a free-for-all. At that point, Jochi and Chagatai may no longer be of any use (they may possibly even be liabilities), and ambitious pretenders may want them eliminated (perhaps one of them escapes with schemes of becoming Genghis 2.0?)
Alternatively, one of GK's brothers could step in as Khan, but it's doubtful that they'd command the same level of control, so expect a schism there too.
As for the Jin, they'd be damaged and shaken, but not, I think, irreparably. They'd mobilize to kick out the feuding Mongols (though you're right in assuming that the Mongols would go back anyway to bury Genghis; tradition demanded it), possibly launch a few punitive expeditions into Mongolia itself (adding to the chaos of the emerging civil wars). In the long run, the Jin would probably be weakened against the Song, so expect the Song to gain some territory, and possibly to be the ones to reunite China down the road.
The Xi Xia may survive for a lot longer, if they play their cards right.
Outside China, the butterflies would be pretty spectacular... The Islamic world is arguably going to be the biggest beneficiary, especially Persia and Mesopotamia (both regions that were utterly devastated by the Mongol conquests). The POD butterflies Timur and the Mughals (though the Mughals were actually a pleasant result of the Mongols in OTL, but I'm sure India will get some other epic empire, probably a larger/more powerful Vijayanagara). Baghdad will avoid being wiped off the map as it was OTL, so likely the Abbasids will remain in place as figureheads for a while longer, and the House of Wisdom will be preserved along with many other of Baghdad's wonders of the day. In fact, a lack of Mongols in Islamic lands IMO drastically reduces the chances for radicalized/fundamentalist Islam to emerge and helps maintain the Islamic world as a global centre of scholarship and innovation (Renaissance Italy will have a formidable rival with a well-established scholarly tradition, in other words). The Ottomans will probably be butterflied, but don't automatically assume that means "surviving Byzantines"; the Byzantine Empire will likely fall at some point, if further down the road, most likely to Serbs or Bulgarians (who would then probably start calling themselves "Romans", at least in a ceremonial sense), less likely to Italians (who probably wouldn't).
Eastern Europe is yet another matter... Russia will probably remain divided into a bunch of smaller states... It could unite under pressure from another outside source (Poles, likely), or remain in that state for centuries until being united in a Germany-like or Italy-like manner in the 19th century.