Alternate history thought experiments are useful for reversing the anesthesia of the familiar - mixing terms, switching coincidences, you start to see where things might have been accidents, and where other things were probably destined. Hence, the "sinization" of European history, combining Europe into a super-state, and the Europizing of China's. But it didn't happen that way. It's still interesting to consider how Europe would seem if shoe-horned into the patters of Chinese history and vice versa.
From a Chinese point of view, Europe is a just a Warring States Period that never ended; a China that never got conquered by one government. Several of the warring states dominate the picture. They are those which have access to coastline and historical political access to new territories, as well as those with good cultural institutions that serve them well in the industrial age; those seem to be states that lacked strong central control until recently, especially those which were not dominated by an empire outside Europe. Europe is a multiethnic continent that has never been truly united. We still remember the Roman empire 15 centuries after its fall because it controlled two-thirds of the continent's territory at one point. Europe is often divided into east-west (although most of the east is now in NATO and this line has seemed to waver in the space of decades); more enduring is the north-south division, created by the more permanent physical barrier of the Alps and their weather effects, which we can see in religious and other cultural divides. Other conquests from without have been averted: the Persians, then the Moors, then the Mongols, then the Turks, all of whom who got pieces of the proximate fringes of Europe. (Don't forget that western Turkey was settled by people we would recognize as Greek and they were ruled by the Persians and fought for Darius at the Battle of Marathon.)
On the other hand, China is just Europe that got conquered, and never re-fractured. Pick your unification point - a loss at the Battle of Salamis, and/or Alexander going west instead of East, or the Roman Empire never falling, or at least always having its territories inherited intact by future rulers. When the Qin took Guangdong (modern Canton), they were culturally and ethnically distinct from the people they met, who they described as bamboo-thicket dwellers barely more advanced than hunter-gatherers. (Many Cantonese people even today who self-identify as Han in fact are more genetically similar to Cambodians or other southeast Asians than they are to Han from the rest of China.)
Miao women in traditional dress, from chinapictures.org.
A glance at the map shows the old kingdoms of East Asia - the old Warring States - and the briefest of research will reveal China's persisting multiple ethnicities - unsurprisingly in its interior (the Uighurs and Tibetans) but even much further east (like the Miao women above, or the recently migrated Hmong, and many others) - and many of these are finally today losing their cultural distinctiveness, long after losing their political autonomy. The parallels in my little essay are fairly obvious. In that world, there is no "China", there's just East Asia, a collection of belligerent ethnically distinct states with confusing alliances, and an unlikely overarching religion (Neo-Shivanism) from a very foreign place that somehow does not cause them to unify. The Persian state was the Qin, who didn't tolerate lots of chattering scholars either; the burning of the library at Alexandria was nothing. The Macedonian and Caesarean dynasties were the Han, and the Norse were the Mongols. In the alternative universe the East Asians have their own barbarians (the Xiongnu who served as mercenaries in the Han legions) and their Scandinavians, the Mongols and Manchus. In the alternate-history Europe, the last of the Huns and Basques are losing their dress and cuisine, while the Russians and Turks remain politically problematic. (Perhaps Vladimir Putin is the Dalai Lamasky in that world? Who knows.) If you find a Norse or Finnish dynasty unlikely, why, that's exactly what the Sung would've said about the Mongols, and the Ming about the Manchus.
So what are the answers to the questions posed by my alternate history alter ego? That is to say: why am I real, a German-descended person with a Hebrew first name living in a United States settled by the British and named after an Italian, and he's fictional? (I'm not debating whether I'm the real one. Don't get all Chuang Tzu on me.)
1. Why did the super-state of China unify so early to be ruled by one dynasty after another, with only brief periods of fracture, in contrast to Europe, which has only been half-ruled by one state and only for a few centuries at that? This one may be straightforward: geography. Most of eastern Asia is a warm fertile plain, and you could walk from Shaanxi to Guangdong, eating rice along the way - which is exactly what many armies have done. (You would expect that places with geographical challenges would resist incorporation. Within China, Shu, now Sichuan, developed a distinct culture, and of course Korea and the tropical-forested Southeast Asian countries were never absorbed.) On the other hand, Europe has an insanely complicated coastline, plains frequently interrupted by mountains, and a shorter growing season. Europe is harder to invade, harder to administer, and until relatively recently has been less of an economic prize than much of the surrounding world. This is why the Romans ruled Syria but not Denmark, why Alexander went to Egypt and not Rome, and why the Moors only half-heartedly pursued Europe beyond the Pyrenees. In particular, the ornate coastline of Greece looks like something an overenthusiastic ten year-old geography nerd would invent.
2. Why did religion in the West evolve such that it was synonymous with political power? Why is religion in the despotic East syncretic and tolerant of other traditions? I do not have a strong answer for this, but here are several speculations: a) a statistical-evolutionary one. More religions evolving means more natural selection, and therefore more that want to spread and hold power (ie, evangelical, politically-aggressive religions.) Therefore, if you're a state in contact with more religions, you're more likely to encounter politically aggressive religions. If you're in Europe this is the case; and the Middle East was also more closely connected to Europe than to China. (This is the same reason in biology that Eurasian/Mediterranean species are more likely to become invasive in the New World and Oceania than the other way around - a larger initial territory means more species competing, and by the time of the Columbian exchange, only the best-adapted ones were left in that Darwinian crucible, and they often overwhelmed the ecosystems they encountered.) b) Large, centralized absolutist rulers who are serious about staying in power do not tolerate flourishing ideas: hence, the purge of philosophers that occurred with the foundation of China. Who knows, maybe if the Mohists were still here today, they would have spawned something as intolerant as the Abrahamic religions? Plus, religions in an absolutist state where the head of state is not interested in religions are not in a position to develop into an ideology that tries to purge other ideologies.
3. Why did the technology and wealth of European states progress rapidly beyond that of China beginning just prior to the age of discovery? Economists and historians have put the most thought into this question, and one thing we do know is that the groundwork was laid well-prior to the age of discovery. That is to say: the answer wasn't just that Europe extractively colonized the world and benefited from the labor of non-Europeans (although that helped.) There is an argument that natural selection built states in Europe that were militarily successful, and that living in a state which was able to provide predictability and safety but was otherwise not authoritarian would tend to select at an individual level for values that would be useful in the late iron and industrial ages.
This immediately relates to current trends. If we are now in an age more dependent on administrative competence than military skill, China has the advantage. But if China's recent successes now come partly from implementing ideas and values the West has stumbled upon through natural selection, a continued strongly central authoritarian China would not be expected to keep producing such. And that by extension, a strong world state would be an absolute disaster in terms of cultural innovation. Ibn Khaldun, the Persian historian and observer of China, made an observation about this, just as relevant to China's initial unification as it is to the source of its continuing dynasties and ultimately the impact on civilization of the habitable world being entirely divided into states, as it now is. Why would the unifier of China be from Shaanxi, which even today is a bit of a backwater? Why did the first European conqueror not arise from Sparta or Athens, but from those upstarts Macedonia? Whether or not he was right, Ibn Khaldun thought there was an effect of outsiders on the frontiers of an empire coming in to take over the complacent inward-gazing capitals, at least in Chinese history, and he had the benefit of many additional dynasties' worth of data to make his argument than just the Qin. He was perhaps over-weighting the significance of the Mongol dynasty in power at the time he was writing. But what if there were no more clever barbarians on the frontiers, but only other states, operating in a cartel? I don't know, but some level of natural selection, and therefore the triumph of good cultural practices over bad, seems very likely to grind to a halt.