The directions of Siberia's river systems rather helped, I believe. The Volga (which branches into Siberia) was a major river in Russia prior to expansion into Siberia and the short portage between the east-west rivers made it fairly easy for Russian Cossacks to get from the Urals to the Pacific in only about half a century.How did Russia steamroll through Siberia while China had trouble?
China, on the other hand, didn't really have much incentive to explore a frozen wasteland (since that's what it was prior to discovery of its mineral resources, hindsight is 20/20). The Amur was only a border of the northern reaches of the Manchu tribes and didn't play too huge of a role for the Chinese empires at any point, so exploring its reaches wasn't too big on anyone's priority list. There was plenty of more temperate land to settle+conquer to the south and millions of Chinese left for Indonesia, the Philippines, Malaysia, Thailand, etc. over the centuries. The money and drive was never in the north, so it's natural that China hadn't really touched the region at any point. Russia, on the other hand, had it as a natural extension of their major river and trade routes, there was more the will to expand that way and acquire its money.
The Han held northern Korea for a few centuries. Plus, the Tang managed to break the Goguryeo by siding with the Silla. However,Tell that to the Sui.
None of the post-Han dynasties managed to hold onto much of the Korean peninsula for long. Rather than it being difficult to hold, it was more like they stopped trying altogether because it was more worthwhile to cooperate than to send troops to a hostile land (the Goryeo suffered 7 invasions by the Mongols rather than bend the knee entirely).Korea wasn’t as difficult to hold.
If the Chinese empires tried, they probably could've subdued the whole peninsula but only after significant costs and causalities, judging by every invasion attempt of the peninsula by outsiders. Not quite on the same level as Vietnam but it'd take a significant campaign to conquer and hold, seeing the result of the Tang-Silla War. Then going ahead to invade Mongolia, Central Asia, Siberia...that's probably not as likely, since the logistics would've been difficult to manage, especially with the need to administer the new territories and suppress any attempts by the Koreans to gain the independence that they were used to. It would've been costlier to do than to simply coexist with tribute. The Chinese empires conquered lands for wealth and border protection. Korea wouldn't have added a huge amount of wealth relative to the cost of invasion and the border wasn't too much of a concern (the Korea kingdoms weren't suicidal; they knew fighting the Chinese would not benefit them either. They could fight but they wouldn't gain anything meaningful from the conflict the way nomads could loot and flee).
It'd take a very rich, very stable empire quite a while and a series of very aggressive empires and weaker bureaucrats and eunuchs to go ahead and conquer all of that and a few good emperors and centuries of luck more to actually keep them permanently. As for effects, Korea's already plenty Sinicized but maybe a bit more Chinese influence in the western regions, just from settlers and increased trade. No Hangul, just Hanja, naturally. Unless the government forced settlement, Tibet, Mongolia, Siberia, and Central Asia probably wouldn't be very populated, with current day Xinjiang and Tibet in mind. Maybe a bit more but there's just better land to live on elsewhere.
And, of course, it makes it easier for regional identities to form and the central government's expenses to rise just with the vast addition of so much land. Warlord eras would be worse, probably, with more (stronger) contenders.