I like where this is going. Here are some thoughts:
1. The way to make the Mongols more of a 'net positive' is mostly to prevent the whole war with the Khwarezmid Empire, so that the Mongol Empire is smaller, but enjoys good relations with thev Khwarezmids. This would help heep the Mogol Empire more cohesive, thus more likely to survive as roughly one entity. That entity, due to its size, would have to be quite decentral and tolerant/cosmopolitan by nature. The huge size of one area allows for a "Pax Mongolica", the good relations with the Khwarezmids allow for peace with that major neighbour, and the decentralism/tolerancy/diversity means that all sorts of creative impulses (and regional competition) can still arise-- which would be beneficial.
2. Preventing the initial casus belli against the Khwarezmid Empire may not be enough to prevent war in the long term. Let us say that the Khwarezmids don't attack the Khan's emmisaries who operate under the terms of a treaty, and if any raiders do so, condemn it and send their heads to the Khan. let us further say that over time, the two empires conclude increasingly firm treaties of commerce and amity, and remain on good terms for at least a decent period of time.
3. For the best results, let us assume that Genghis Khan, rather than mostly planning yet other campaigns, spends the period of the OTL destruction of the Khwarezmid Empire by consolidating his Empire as best as he can, furthering ints cohesion and the prospects for the retention of its unity in the future.
4. Even if not spending time and energy conquering the Khwarezmid Empire, the oft-seen-in-AH "Mongol conquest of Europe" will turn out to be impossible. The Mongol armies are not suited to European warfare, and even if they have more resources to throw at it... the logistics will simply make it self-evident that Europe is "more trouble than it's worth". I'm sure the Mongols, if fully dedicated to it at the height of their power (with no distractions), could probably make a lot of inroads. No doubt about that. Even if they were unsuited to the terrain, their were astoundingly flexible in their strategy, and could adept. (See: Mughal Dynasty.) But the fact is, the Mongols weren't stupid, and it just wouldn't be worth it.
5. The above means, however, that the attempted invasion of Europe probably occurs slightly earlier. That, plus the ATL additional steps by Genghis Khan to get his house in order, means there's no Mongol in-fighting (as in OTL c. 1248) to make things messy. Instead, we see a controlled withdrawal and a deliberate decision not to pursue European conquest any further.
6. Again contrary to "bigger is better", it's best of the Mongols do not waste time attempting to annex Southern China. Optimal for the Mongol Empire is if it does retain Northern China, thus becoming a power that essentially controls all of Northern Asia. As with the Khwarezmid Empire, the best situation is lasting peace with the Chinese state to the South-East.
7. Presumably, the focus of the Empire, even without Southern China, will become increasingly Chinese. We may assume that the bits that essentially correspond to European Russia in OTL will ultimately split off. If the Mongol Empire actually stays together as one loosely-controlled polity, this may very well happen earlier. (It's easier to split off from the very-far-away ruler of a vast empire than to get out from under the thumb of the far closer despot of your region's successer state to said empire.)
8. This means that "Mongolia" will essentially control everything East of the Urals, North of OTL Kazakhstan and (east of that region) North of India, as well as most of China (just not the South-Eastern part). So it's going to be an Asia-spanning entity, much like OTL Russia, except coming from the East instead of the West. Assuming that this empire retains the general policy of tolerance and diversity-in-cosmopolitan-universalism, we could be looking at a more world power here. And quite possibly one that will be very receptive to technological and scientific innovation! That's not to mention the safety that the imperial roads would afford to travellers (assuming this remains a priority), which would greatly foster trade.
9. Korea will presumably be annexed as well, although I'm not sure Japan is sensible as a long-term prospect. Even if the Mongols try and, in this AT, succeed in conquering it... I thing their Empire will be oriented towards the continental trade. Some insular possessions are probably going to be marginal, and there's a good chance they get spun off into one or more increasingly autonomous vassal kingdoms before long.
10. Final note: the POD, namely that the Khwarezmid Empire isn't conquered, has an additional effect that can really help. Far fewer Muslims in the Empire. No, I'm not saying Muslims suck-- I'm saying the Mongols, if they retained so many of them, would very likely end up becoming a Muslim power themselves. Like the Timurids and the Mughals. It's actually better if that doesn't happen, and the deliberate policy of "all religions are tolerated so long as they don't force their shit on others, and the state has no formal religion" is retained. Genghis Khan was way ahead of the curve there, and it would be best to retain that. It would simply be anoher factor that could help give these ATL Mongols a cool advantage.