Assuming that there is no Imjin War and Ming China and Joseon are a lot stronger in this time period, but they're too busy with some internal matters, how about the Manchus? I'm not sure if Siberia could be colonized by any remnant of any Turco-Mongol khanate or something. The only way for a Turco-Mongol Siberia to be established would be to cripple Muscovy permanently.
If there was no Imjin War, then Joseon and the Ming probably would have had more influence in Manchuria. The Manchus, which had originally been Jurchen, were not unified until the early 17th century, and both Korea and China would have attempted to keep the area divided in order to continuously receive tribute. Also, Gwanghaegun, the Joseon ruler soon after the Imjin War, was a very skilled diplomat and had been reorganizing the military before he was ousted due to political reasons. Regardless of whether Japan invaded or not, if the government had supported him more firmly due to butterflies, then he might have attempted to expand into Manchuria.
People also need motive. Staying home might be rough, but at least you are in a world you know. Going to a new place is a risk even if it is a sensible one to take.
For all we know, unrecorded dissent elements of Chinese, Korean and Japanese societies mighthave taken ship to Siberia and set up a colony, which went to the wall. The Europeans often only made it in North America because the natives bailed them out. There are a fair few cases where they did not.
All we know is that the governments did not think it worth their while to attempt to set one up. If Norway could maintain suzertainty over Greenland and Spain govern Mexico and Peru via viceroys, China could have easily governed a colony via mandarins and Korea and Japan equally as well. I reckon they did not because within the values they held they saw no good reason to do so. Whatever Ssiberia might hold, it could be obtained much easier closer to home.
As I said before, China, Japan, and Korea were occupied with other events to think about Siberia. After 1000, China and Korea were continuously occupied with the Khitan, Jurchen, Mongols, and the Manchu, while Japan was concerned with its own affairs to think about looking beyond Hokkaido. Dissidents might have traveled to Siberia by sea, but unless there were significant tangible benefits to living there, it would be very hard to convince the government.
Not quite, the Qing actually controlled Southern Siberia and ventured, briefly, there as well. If there was any sort of serious competition between a Chinese Dynasty and a non-Chinese Dynasty that they did not merely view it as a Barbarian State wanting Barbarian Lands, such as perhaps between the Sassanids and Tang (Though that is a far older example).
For Japan... please look up the Daimyo of the Kakizaki-Matsumae clan. Now Hokkaido was never fully colonized until the Meiji era, but this stemmed from a cultural belief in the unnecessary and barbarian nature of the north. Should Japan ever develop the desire to expand instead of simply tolerating their existence with a Japanese march with very few troops to keep them in check expansion beyond Hokkaido would be pretty plausible.
Korea is not so lucky, needing to conquer both the various tribes of Manchuria, resist the powerful Chinese, and either subjugate or conquer those of the Mongolian Steppe. In a word, suggesting the French colonize Siberia by way of Germany, Poland, and Russia.
The Qing was originally a Manchu dynasty that was eventually assimilated into Chinese culture around the mid-18th century, and significant migration to Manchuria did not occur until the 19th century when the Qing allowed the Han Chinese to relocate in response to Russian incursion. Until then, the Manchus had planned on closing off Manchuria to settlement in case of a Han Chinese resurgence that would force the Qing out of China proper. Meanwhile, the Qing was more concerned with states elsewhere, conquering Xinjiang and Tibet, while invading Nepal, Burma, and Siam. Also, it's hard to say whether the Manchus expanded into "Siberia," because a current Chinese interpretation is that the northeastern areas that they had lost to Russia should be considered "Inner Manchuria," and neither Manchuria nor Siberia have concrete definitions. In other words, even if the Qing attempted to expand into Siberia, it would be hard for them to actively colonize the area as they were not focused on mass settlement at the time.
I see your point concerning the Japanese influence in Hokkaido, but the Imjin War caused Japan to withdraw into isolation, as it had suffered numerous casualties after two costly invasions, and probably would not have attempted to actively expand soon after. One of the reasons for the invasion was that Japan supposedly (and incorrectly) assumed that the southern Korean Peninsula had been under Japanese control, not to mention that Japan sought Korea as a gateway in its attempt to expand into China. As a result, I think that it would be hard to butterfly the war away if Japan had been militarily unified by the late 16th/early 17th century, and if it remained divided, then none of the states would most likely have had the capability to expand into Hokkaido. Attempting to reach Siberia after the mid-17th century is unlikely due to Russian excursion.
Meanwhile, Korea had slight, although unlikely possibilities, in attempting to expand into Manchuria after 1000. We already established on another thread that in the early 12th century, Goryeo could have potentially seized significant areas of southern Manchuria if it had managed to maneuver the tribes in Manchuria. Also, in line with the above response that I made to MarshalBraginsky, Gwanghaegun or another skilled diplomat/military leader could have potentially done so as well.