Well, China was divided into many states from 800 to 200 BC, about 600 years. They did not reunify quickly at all. If something similar were to happen, say between 1300 and 1600, you could have well-developed coastal kingdoms (like a modern Qi state) that is forced to maintain naval developments in order to offset other disadvantages.
I know:
Yes, but China's history didn't begin in 500 AD. China was divided from around 800 to 221 BC, then again from 190/220 to 589 AD, with a brief exception from 280 to 304 under the Sima Jin, so that's roughly 950+ years of division, out of about 1400 years, from 800 BC to 589 AD. However, it was then divided again during 907-960, and lastly from 1125 to 1279, when the Mongols unified China under the Yuan in 1279, and China did generally remain united until the Qing fell in 1911.
I did edit that post after you posted, but it was because I misread 1500 as 2500, and I did not edit the specific dates except for adding the fall of the Qing in 1911.
However, the problem is that South China was much less densely populated in comparison to North China until the Song or so, and the coastal population eventually began to outnumber the inland population since then. As a result, you're talking about two completely different scenarios here, and even more so because a state in the 14th or 15th century roughly covering the areas that the Qin had controlled would almost certainly be unable to resist the other states in China Proper, let alone conquering them. Again, as I stated earlier, states that are located along the coast, or along rivers, will be able to conquer those further inland, and if a balance of power is not maintained between 2-5 states, then one will probably end up unifying China.
Anyway, in terms of Japan, I will focus mainly on the west, north, and south, as I think that going east without going north or south first is virtually impossible.
Regarding west:
Korea managed to fight back with around the same number of troops, if you combine the independent individual armies put together in response to the invasion. Because Korea had resisted numerous military incursions from outsiders for hundreds of years, and generally succeeding, with the possible exception of agreeing to become a Mongol vassal, it would have been extremely unreasonable for the people to merely sit on their hands after the Japanese landed. Due to resistance in numerous villages/cities often operating independently of the government, not to mention the navy, some technological advancements, such as the hwacha, and token support (in terms of manpower) from the Ming, Japan was forced to abandon Korea altogether after eight years.
It's also extremely telling that Korea reverted back to its Confucian method of thought after the war, even though that was the main reason why the army had become decentralized, causing the Japanese to sweep though the peninsula within a year, and that Japan generally pursued a policy of isolationism for more than two centuries until it was opened up by the West. Of course, both states eventually resumed trade with each other and other ones, but the trade volume never reached a point comparable to that of the Europeans. In other words, generally speaking, because the states in East Asia were content with the resources they possessed and the trade that they conducted, there was no particular reason for them to look further outward, either economically or militarily.
[M]ilitarily speaking, Japan's ultimate disaster in Korea illustrated how it was unable to expand outside of its borders before the late 19th century, and economically speaking, the state was generally content in terms of trade with East Asia and minimal trade with the West, mainly through the Dutch. In other words, given the general situation at the time, it would be extremely difficult for Japan to create more trading routes with SE Asia and the Americas, and possibly expand militarily outside of the islands before the mid-18th century, initially through Korea.
In addition, I think people are generally more focused on how Japan managed to raise and arm a large number of troops in a short amount of time during the Imjin War, but they overlook the fact that Korea's population was only 1/2 to 1/3 that of Japan at the time, and yet it still managed to raise a similar amount of troops. Also, Joseon lacked western firearms, although the range for the Korean bows often diluted the Japanese advantage.
Regarding north:
I was talking about a divergence shortly after the Senkoku period began . . . foreign influence would probably not be butterflied away, and if the Europeans were equally intrusive, Japan would probably also be xenophobic no matter who was in control at the time . . . if Japan decided to look north, then west from Hokkaido and potentially Sakhalin, I guess it would be possible, although it would probably be limited to trade with tribes in Siberia rather than actual takeover or maintaining colonies. The gradual expansion would also take decades or centuries, allowing Russia to step in by that time.
I see your points concerning an alternate Japanese regime. However, the Japanese probably decided to conquer the Ryukyu because the islands had become profitable through trade, while taking over Hokkaido would probably not have led to tangible benefits. Theoretically, the Japanese could expand into Sakhalin, and establish trading relationships with tribes in Siberia, as I stated before, but there would really be no impetus to colonize or actively expand into Siberia.
In terms of expanding south, taking Taiwan would almost certainly have received a direct response from China, and the Philippines were firmly under Spanish control by the time that Toyotomi Hideyoshi came to power, so although some trade links could be established before then, a military campaign would almost certainly be disastrous.
Although this is not directly related, in terms of Korea:
If there was no Imjin War, then Joseon would have most likely allied with the Ming against the Manchus, then Russia. The result would be a remote possibility of Korea initially gaining portions of southern/eastern Manchuria, then expanding into what is now Primorsky Krai, and possibly reaching the Sea of Okhotsk. Because Joseon would have a population of about 10-15 million by the mid-17th century and around 15-20 by the end of the century after settlement and growth, the initial amount that would have been settled in Manchuria/Siberia would be about 1-3 million. For comparison, tens of thousands of Joseon farmers settled in Manchuria although they were technically forbidden from doing so by the Qing. However, like the other scenarios, the possibility that all of the events would fall into place is very unlikely.
In other words, although it is possible that China, Korea, and/or Japan could have expanded further and/or established colonies, it would be extremely unlikely for the governments to actively sponsor the expeditions, which would make it almost impossible for widespread settlement to occur.