In particular, how did Imperial China and later Republic of China manage to achieve unity across such a large and diverse geographical area? What factors allow a single nation state to endure at such immense scale, while others fail to achieve stability and cohesion despite being much smaller?
I think much of it's about variation of early settlement density within regions: a lot of people in one area can expand into an undeveloped or sparsely-populated neighbouring region without too much cost to themselves. China was able to settle its south with a largely homogeneous northern population, likewise medieval Russia its east (and south), and northern India its east (but not the south) - and of course the 19th-century US enjoyed large numbers in the east, far fewer to the west. In contrast the Balkans and the Baltic states found themselves on or close to migratory thoroughfares or invasion routes, giving them a mix of populations (though the same can be said of much of Russia) and putting them at the centre of others' conflicts, which impeded modern state formation: in the case of the Balkans, greater powers later deliberately obstructed the emergence of large states that might challenge their influence.
If you want to be really big in terms of numbers, being a subcontinent with natural barriers helps: we think of India, but I think China answers the description too - in contrast Russia's main barrier's plonked in the middle, and Russia today covers only a part of area of the former Empire or USSR, other parts of which merge into wider geographical or cultural regions.
But perhaps the underlying lesson is "Don't have over-powerful neighbours".