How does the world change if the Mongol Empire either doesn't form or remains a very limited size?
Western historiography has largely overstated this. Fact is there was no single "intellectual centre" in the Islamic world, when we have other centres and we also see Islamic learning thriving after Baghdad's destruction, even under the Mongols, despite Baghdad losing its prestige.The Mongols destroyed the intellectual center of the Islamic world
Gunpowder isn't what made Europe dominant, naval power was. This is especially the case because non-Europeans also had gunpowder technology and only until the 19th century can we decisively say Europeans surpassed all others in gunpowder weaponry, by which time they had already conquered most of the Americas, much of Oceania, most of India and not insignificant portions of South Asia and Africa.All of these are believed to have contributed to European dominance. Removing them may butterfly this development away.
Again, this is a Western historiographical nonsensical myth that ignores that Baghdad wasn't the only Islamic centre of knowledge, when you had the likes of Timbuktu and Cairo that were never touched by the Mongols, and that ignores continued Muslim development of technology and science and continued prosperity. Some of the most beautiful Islamic art comes after the Mongol invasions. Look at Uzbek and Ottoman mosques for example. The Ottomans for that matter were masters of gunpowder weaponry and were sponsoring overseas expansion, including as far as the Horn of Africa with the Ottomans even having a province there. Heck, it is after the Mongol period that we see Muslim seafarers take an even bigger interest in Indonesia and the Philippines. It's after the Mongols that Indonesia becomes predominantly Muslim rather than just being a hub for Muslim traders with only a few Muslim states if any. The Mughals are also noted to have produced almost 1/4th of the entire world's GDP at their peak. Again, this idea of Mongols largely killing Muslim culture is a myth that needs to die and die hard at that.The Islamic Golden Age continues unabated and there are probably more advances than in our timeline. The Islamic Golden Age came crashing down after Mohammad II of Khwarazim ensured the destruction of his empire by killing Genghis's envoys. After the destruction of the Khwarazim Empire, it was open season on the Middle East, resulting in the fall of Baghdad in 1258.
Europe wasn't "backwards" nor were other regions less feudal than Europe. For starters, Europeans had arguably far better seafarers than anyone in the recognised civilisations of Asia and Africa. The Vikings reached the Americas by ship before Muslims or before the Chinese despite the latter being relatively close to Canada and Alaska. Viking clinker built ships were the same ships used by Crusaders to invade and conquer portions of Syria-Palestine that eventually led to the conquest of the Eastern Roman Empire itself and the first takeover of Constantinople. Medieval Europeans also had windmills and water mills that didn't exist before in the continent and which were no less "advanced" than what was in Asia. Some of the best steel in the world was in Europe. The population density was inferior to India and China, but not to that of the Islamic world, and Europe always had an inferior population density to South and East Asia anyway even during the Greco-Roman period. The myth of early medieval European backwardness compared to other parts of the world also needs to die a violent death. Also, I really doubt the Black Death had anything to do with the Reformation when the Reformation started nearly two centuries later and even the proto-Protestant Jan Huss emerged several decades after the Black Death, while we have earlier heretical movements like the Waldesians and Cathars, all while the authority of the Catholic Church was always contested by powerful monarchs from France and the HRE and even England.The Black Death doesn't happen, since by reviving the Silk Road, the Mongols opened a path for an Asian disease to make it's way to Europe and also, the Mongols often used plague-ridden bodies as projectiles in their sieges, so any refugees that came from newly-conquered Mongol territory would be carrying the plague. No Black Death means Europe maintains a large enough population to ensure he survival of feudalism. In our timeline, in less than 3 years, 1/3rd of the European population died. Europe would remain backwards in terms of social and political progress for at least a few more decades. Also, no Protestant Reformation since the seeds of doubt in the Catholic Church's supremacy were born during the Black Death.
The type of printing press you speak of was developed independently in Europe in the 15th century. The lack of a Mongol Empire doesn't stop this development.Without the Yuan Dynasty, the perfection of movable type printing by Wang Zhen probably never happens. Without the printing press, information becomes harder to spread around and is more easily cotnrolled. Previously, movable type printing had existed since the 11th Century, but it wasn't until the 13th Century it was perfected. I don't doubt that someone in this alternate timeline could perfect the printing press like Wang did, but if it's not invented when it's supposed to be invented in OTL, the effects are completely different.
What is "proper agriculture"? People in Mongolia and the East Asian steppe have been growing crops for centuries. And why do we always speak of "isolation" in regards to China when China already under the Tang was an empire, when counting vassals and tributaries, of about 10 million square kilometres and receiving traders from as far as Iraq. There were Africans in the Tang Empire for Christ'sake. It also had different religious minorities, including Jews, Manichaeans, Nestorian Christians and Tengriist pagans. It hosted the last Sassanid monarchs as if that wasn't enough. Already the Han had contacts with Persians, Greeks and Romans, and also knew of Malaysians and their seafaring too. China has never been more isolated than any other culture in history.Without the Silk Road to provide a market for Chinese goods, China is worse off in this timeline. Different ruling dynasties aside, without Kublal Khan, animal husbandry remains the norm, with no movement towards proper agriuclture. The development of the textile industry could remain as it is in our timeline, since Huang Daopo didn't seem to rely on any government support for her work with cotton textiles. In regards to trade, China would probably remain isolated for longer than in our timeline, with no push towards opening ports to receive foreign goods such as spice and medicine.