Okay, another one of these threads.
One: the Japanese navy and naval tactics were suited more for its local seas and were largely boarding actions, so unless you are radically changing how the Japanese approach ships and sea combat within a generation, that's a problem. Two: why does Korea let Hideyoshi use their lands, that leaves with us two scenarios, Korea caves into Hideyoshi or he somehow wins the inevitable Imjin War which he lost because of number one. Three: What does Ming look like at the time of a Japanese invasion, the only way I can see anything going Japan's way is the if the Jiajeng Emperor's reign goes completely and utterly south, but that requires a POD that can butterfly away Hideyoshi's very unlikely rise quite easily.
Even if we assume Hideyoshi somehow wins out, what does this China even look like? Japan was run by warrior aristocrats and local strongmen where the power in Japan, Hideyoshi did unite the country but still had to worry about the fact he was on shaky ground legitimacy wise. Hideyoshi was a peasant who could not become Shogun himself and had succession troubles with the Toyotomi clan. China, on the other hand, was a bureaucracy based on a mix merit and having some powerful friends or the Emperor's attention, where the military could at least be kept in check. You couldn't have two different systems to try and reconcile.
So if Hideyoshi could conquer Japan, the first question is what happens to the Chinese concept of Emperorship? Hideyoshi trying to set up himself as emperor would be out and out political suicide, the locals might not like him, and would probably piss off any allies he had in Japan So there's always the possibility making the Japanese Emperor, Emperor of China as well.
Again who becomes the administrators of a Japanese controlled unless you get a Shogunate system in China that is ironically enough closer to its roots as a bureaucratic state under the Emperor, with various posts, than what happened in Japan, or does feudalism come to China, and do you get Chinese Daimyo. There's also the issue that the Japanese Imperial Court had only begun to recover from having it's Emperor's selling their own artworks for coronation money, would Hideyoshi really want to risk making the Imperial Court more powerful. Second, how do you reconcile the culture of both courts, while the non-Chinese dynasties did adopt harems and eunuchs, it would be different, as Japan had its own Imperial Court with its own customs.
As for colonialism, I don't see it happening Japan was still decentralized, that any idea of resource exploitative colonialism would be out and out impossible. The clans that might bother to care are the clans in western Japan, and they would just want to trade with China and Korea without conflict.