I too like the projection. The geography seems fine enough.
The political boundaries confuse me a bit. I understand things like combining the Russian principalities, the alliance of the Georgians and Armenians to the Byzantines, and compacting the Ayyubids and Abbasids - all for game purposes. But the apportionments of Italy, Germania, and the Levantine crusader kingdoms are odd to me. I'm assuming that the map represents the dawn of the 13th century, or not long before that. So why is northern Italy not part of the German Empire and the south of Italy the Kingdom of Sicily (Normans?). Also, if possible, you might separate some of the Turkic polities (Seljuks of Anatolia, Bulgars and Cumans of Volga-Don, etc.)
I can't speak with much knowledge regarding the east Asian cities, but the European and Near Eastern cities seem off as far as population. I don't know what size town you're representing with the various numbers, but I'll give an example: Kiev and Venice are both size 6 cities, yet Kiev would have been smaller (less than 15,000 population) compared to Venice (somewhere in order of 25-50,000 people in this era). In the triangle between Ghent/Bruges (which are not on your map), Venice, and Novgorod, there would have been no towns of significant population (above 15,000). Likewise between Constantinople and Venice.
The corridor between Milan and Venice was heavily populated (Brescia, Verona, Padua), and the absence of Pisa and Florence omits some populous and influential cities that contributed to the heavy population of northern Italy.
Between Baghdad and Basra was Wasit. Near Cairo and Alexandria was Mahalla. Jerusalem was nowhere near the size of Antioch. Qus, up the Nile from Cairo, is missing (about where you've named Luxor).
There's just a bunch of cities on the western side of your map that shouldn't be near as big as some others.
You've not placed resources or barbarians, so all I'll say on that subject is to suggest many barbarians in Siberia and in the heart of Arabia - representing Tungus, et al, and Bedouin, respectively.
Lastly, it looks like there's wrap around, so the Japanese and Europeans are short boat ride away.