But Japan was quite lucky when it came to Korea. The Joseon King Seonjo ignored the talk of Japan's unification and militarization and did not prepare the country at all for a war (forts, navy, military buildup) and tried to imprison Admiral Yi, who was rather successful in keeping the Japanese from resupplying, because he was afraid of a coup or some nonsense rather than his country being conquered. He also ignored new military technology like the arquebus because he looked down on the Japanese and Europeans. Hence the early Japanese success in the war, with them taking everything up to Pyongyang before the second year of war had passed. And Korea did end up becoming fairly militarized (had lots of firearms in the army) in the aftermath of the Manchu invasions (albeit too late, since the Qing had taken much of China when Hyojong took power and had his plans for a Manchuria campaign) within 50 years of the Japanese invasion. They were apparently quite good shots, I remember reading somewhere.
A more wary king would've kept tabs on Japan and prepared for war, even asked the Manchu for assistance (Nurhaci OTL did offer help, though most probably to gain intel on Joseon and Ming forces for his future plans).
As for conquering the Ming, the Japanese OTL did not endear themselves to the conquered (you know, burning and pillaging, kidnapping entire villages of artisans and looting, and taking ears and noses as trophies, and raping and kidnapping tends to do the opposite), hence the intense guerilla warfare from the Korean side. I don't imagine the Chinese would be fond of that either (the Manchu succeeded largely because of how many Han defectors there were).
They also need to fight their way through the Manchu, then to whomever controls Beijing, then manage that whole territory while fighting South China while occupying everything north of the Yangtze. If they keep up with the looting, raping, kidnapping style of conquest, that's going to be expensive, both in manpower and money. It'll be hard to hold that too, plus how much power gets divvied up (whoever controls the mainland holding would be way too powerful for the shogun to control). Also Sinicization, since those who conquer China tend to end up being conquered by Chinese culture.
Short term? Lots and lots of death. Long term? Eh, depends on how you do it. Lots of resentment, either way, but that's not new at all.