This leaves a heavily weakened Jing Kingdom, and a stronger Southern Song Kingdom which will continue to be separate, this division will eventually become permanent. The South will cultivate an outward outlook in South East Asia. The North will continue as an impoverished state ruined by a long struggle against the Mongols. I’m not yet sure if the North should be taken over by the Mongols or just left as a state to be perennially overrun by Steppe nomads.
The Jin could not really survive this; generally speaking North China was pretty prone to agricultural civilisation collapsing due to steppe nomad pressure, and to steppe nomads then overtaking it and gradually settling down and becoming semi-assimilated.
I predict that the Jin would drag on for a while, then collapse; the Mongol tribes would then take over the Jin territories, likely becoming politically united in the process if not earlier, and will develop a Sinified, but distinctly Mongolian (and likely Buddhist) civilisation uniting both Mongolia and Northern China.
I agree that the Song won't go back north, though.
A Song Kingdom with a more international outlook will improve the overall level of development in South East Asia, which will have a knock on effect in terms of Song trade links and the effects of trade therein. Of particular interest would be the increased traffic along the sea routes, going through India, and into the Middle East and finally Europe.
With no Mongols the Black Death might never each Europe Feudalism might continue on for some time significantly impairing Europe’s growth. This will also have knock on effects; the Church’s position of dominance is likely to never be shaken and will more or less continue on with the course it was on before the Plague. It’s plausible that the whole edifice might come crashing down later if it doesn’t make a serious attempt at reform.
Actually, greater maritime trade=earlier Black Death going by a slightly different path. Italy is going to suffer.
As a sidenote, I would have to say that the Black Death was more of a catalyst than anything else. Sure, it greatly influenced the exact specific way in which things worked out, but Europe was evolving just fine without it. The Church suffered from entirely internal problems as well as from the rise of the cities which was hampered by the Plague, although the Plague and its social aftereffects certainly gave a boost to heresy and mysticism in general. Classical feudalism was dying away, too, or at least its economic and political significance was. If anything, the Plague set that back as well.
It’s probable that the Song will steadily come to control South East Asia more or less. The desired end result is a patchwork of vassals, tributaries, entre-pots, territories and colonies which give them a loose control over the region and its major players. Australia would also be settled by the Maccasans partly because they are awesome, and partly because they might actually bother to if the Song begin actively trading. There will be a general trend towards earlier advancement as trade and knowledge disperse around the region. The Song Kingdom will also acquire territories in India to give it a more force projection.
You are seriously overestimating the Song Dynasty, and China's capacity for colonial and commercial expansion in general. I agree that the Chinese would've dominated the sea trade, possibly even farther into the Indian Ocean (though they would've gotten a lot of serious competition), but even then there is a question of whether those Chinese merchants would even be associated with the Song in the first place. It seems likely that a more powerful Chinese diaspora would emerge instead, especially when the Song Dynasty inevitably collapses (Chinese dynasties always do: corruption grows with every generation, and purging it is a short-term solution at best, whereas at worst it either makes the problem worse by elevating a new, greedier and more desperate generation of corrupt bureaucrats or by simply destroying the state entirely), but even before then.
A tributary system seems inevitable, though it would likely be the same system as existed earlier, i.e. it would be mostly just nominal overlordship over all of civilised South-East Asia, backed up by Chinese colonies established and defended by treaty. Maybe if there is a Chinese
military revival there could be some interventions against those who defy China too openly or quarrel with the Chinese populated free (possibly even a bit like the Medieval European communes, though ofcourse much more Chinese and Confucian) cities, but that's as much as I can see happening here. I agree that it would revitalise trade and exploration and everything else. India, though? Again, at most there are just going to be trader colonies and the Chinese fleet might be there to protect them against whoever dares mess with the children of the Yellow Emperor; nothing more. It's way too far away, and the Chinese never were eager to expand somewhere far away if they could avoid it: and there many, many ways to avoid it.
The lack of a Mongol invasion is also going to influence the Khwarezmian Empire which
was falling apart at the seams if memory serves. Besides, I'm sure that there still will be a Mongol invasion, of a sorts; steppe dynamics have already been firmly established: whenever population or overgrazing or whatever reaches a certain limit, peoples from the east begin to migrate into northern China and Central Asia, and the same will happen here even without Genghis. It might not amount to as much, ofcourse, but it and the various feudal struggles will still easily bring down Khwarezm.
Ofcourse, without Genghis Khan Central Asia would be much better off, what with its irrigation systems intact and such; I suspect that a new empire might rise here, or even several, likely led by Mongol or other steppe nomads but based on the large cities, but with the decline of the Silk Route the Central Asian civilisation will once again decline and the region will become a backwater, either divided between small local states or a part of Iran, the latter being more likely. Iran, meanwhile, will prosper no matter who rules it (i.e. Turks or natives or any more farfetched options like [insert your favourite Ismailite sect here] or [insert your favourite obscure Middle Eastern ethnicity here] or some version of the Mamlukes) due to the maritime trade, and will likely become the foundation of a great empire, hopefully with a capital (and a Chinese trader colony?) in Shiraz. Minimum conquests are likewise-not-screwed-Iraq and Central Asia, but Asia Minor, the Levant and Oman beckon as well, and ofcourse there is East Africa, though any effectual authority there would be difficult to maintain: still, a Shah-en-Shan could dream, right?
Not sure about your fancy Egyptians, it seems to me that without the Mongols to exterminate their opposition and give them a good light they won't really be as powerful a presence in the Middle East, especially since the Venetians and the Genoese are going to be extra motivated to destroy or subvert them. The Venetians are more likely to support a weaker Egypt, ofcourse, whereas the Genoese are going to help any nice Frenchmen who would want to conquer the place. It might work out; a lot is up to chance here, ofcourse.
An interesting side effect might the resuscitation of the Abbasaids as a viable force, fuelled on an influx of excise duties; it would be unlikely but possible.
Nah.
The New World is likely to be a tougher nut to crack, there won’t be as large a technological gap, and the New World civilizations would probably still fall but it wouldn’t fall as hard or as fast and might break away from its European colonial masters in some places. Come on Knights in the America’s would be amusing to say the least.
I suspect that the New World would be discovered much later than in OTL if maritime trade in the Indian Ocean prospers, though a lot depends on how the Middle East works out. Perhaps Egypto-Persian resistance to Italian infiltration of the Indian Ocean might cause an earlier attempt to literally get around this problem, though (but it won't be much good for this if they just go around Africa).
And what happens there depends a lot on who discovers it when, as well as on sheer chance. That said, diseases would still kill most everyone and realistically I doubt that they would recover in time (it's not so much that it is so very time-consuming a process, historically speaking; but what colonial power is
that lenient? The best case situation is ofcourse if North America (or maybe Brazil?) is still discovered first and Central America and Peru are not reached until a few generations later).
The various “Russian” states are also going to be interesting, they will have breathing space to expand and incorporate the steepes into its territory. Perhaps more interestingly Hungary and Poland will be significantly more powerful. The balance of power might well shift to the East in Europe, as the Eastern states continue to grow. The Romani are unlikely to collapse in this either, they might continue to survive and perhaps regain some measure of power.
First of all, the Byzantines were in agony and I am not going to be convinced otherwise.
I agree that Eastern Europe would be in a stronger position, ofcourse. Hungary might keep its own dynasty and expand into the Balkans and Galicia; though Poland was perhaps served by the Mongol invasion in that it was reunited this much faster for it - still, it will be much better off economically and socially when it
is united. Probably no Poland-Lithuania, though. Lithuania in general will be weaker, ofcourse, what with most of its neighbours not being pounded unconscious and all, though it still would have its opportunities, especially in Polotsk. The Teutonic Order was probably not affected too much by all of these things, though.
The situation in Russia was complicated. I don't think there will be much expansion into the steppe, though; the only ones really interested in doing so would be the Chernigovian princes, and Chernigov was very much in decline by then. Galicia and Vladimir are where its at, ofcourse, as well as possibly Smolensk, though the latter was more of a subregional power unless it could grab and hold on to Kiev and Novgorod (easier than it may seem, but also less valuable now than a century ago). Novgorod would keep a lot of autonomy no matter what, though. Chernigov might yet pull itself together, but not until later. I think that both Galicia and Vladimir could potentially united Russia somewhere in the 14th century; Smolensk or Chernigov are less likely; it is entirely likely that Russia will remain a hodge-podge of small principalities and four or five main powers with whom everyone is aligned, too, though this system would probably collapse by early 16th century at latest. Galicia or maybe Chernigov could potentially unite just the south at some point, in which case it is going to be heavily aligned with either the Catholic Church or what's left of Byzantium (actually, okay, there is one way in which the Byzantine Empire might be revived: if it somehow enters a personal union with Galicia. Then Emperor Roman the Russian will draw upon his resources to beat up Latins and Bulgarians, and move the center of his power to the very awesome, the very-not-encumbered-by-uppity-boyars-but-full-of-well-mannered-capital-nobles-and-court-officials Tsargrad); Vladimir might take over the north and the center; Smolensk might unify the central regions, as already suggested, and then act more or less as the French royal domain vis-a-vis the large feudals, though here they are much larger and more independent, like two or three Burgundies at once, but on the other hand no England, so it might work out. Anyway, the surviving urban society will certainly be nice as far as economic and other development goes.
EDIT: Come to think of it, the decline of the Silk Route and the rise of sea trade would actually change a lot more in Europe itself as well. In particular, it would enrich Constantinople again even after the Fourth Crusade; I doubt that the Byzantines (or the Latins) themselves will be able to use this in any decisive way, though - rather, it would make them a this much more appealing target. Crusaders might sack the city a second time, for one thing; the Bulgarians would do their darndest, too. And ofcourse the already suggested South Russian option... though since this also means the revival of the path from the Varangians to the Greeks it makes the Russian situation this much more complicated, since that path was pretty much the vertebral column of the old Kievan Rus, and would favour whoever controls Kiev and Novgorod (and Novgorod is usually
very receptive towards whoever controls Kiev - or was before this path had declined...) for a much more likely and easy reunification of Russia by the early 15th century or so, possibly earlier depending partly on when and how fast this new shift in trade routes occurs. Now, what this means for Russo-Byzantine relations could be anything: Smolensk and even moreso Vladimir are going to be relatively indifferent to Byzantine politics, though the former in particular would still probably take offense to any damage to revived Russian trade there, and there might be a struggle between the Grand Prince of Russia and the Italian city-states and their colonies in the Black Sea region, as well as a lot of political intrigue in Constantinople (but the Russians are unlikely to try and seize power there). The southern princes, however, do have fair enough a chance to seize power, especially since any revival of southern trade would logically benefit them first, and if Russia is united from the south then it might as well end up being Dostoyevsky's nightmare in that the prince might then try and seize Constantinople as suggested above, eventually turning Russia into a backwater province of the Rurikoian Roman Empire.