What would East Asia be like today if Toyotomi Hideyoshi had been successful in his invasion of Korea and subsequent invasion of China?
I see a few possibilities here.
1) The Japanese take over China but are unable to stabilize it in time to prevent the Manchurians from sweeping in more or less on schedule. The short-lived Toyotomi dynasty is a footnote in history books.
2) The Japanese take over China and get enough of a handle on things to prevent further invasions. The Toyotomi dynasty stays in power for a few centuries, and like other foreign dynasties is slowly Sinicized. However, the Japanese plans for further global conquest would be bogged down by the burden of administering China and eventually be forgotten. There would be no strong reason to reduce Ming policies closing China off from the outside world, though, so the Toyotomi dynasty would take much the same trajectory as the Qing. The tributary relationship between China and Japan is complicated and difficult, but they eventually fall away from one another, and western powers once again push their way in.
3) Toyotomi gets it done. His plan was to do what he'd done in Japan: coopt some local elites to help defeat other enemies and build an increasingly powerful empire around them by performing the process again in the next territory over; lather, rinse, repeat. With Japan secure, he takes Korea. With Korea locked down, he takes China. With China in hand, he moves on. Southeast and Central Asia are the next targets for expansion. There's still a time limit, though. The Toyotomi dynasty builds an empire which rivals the Mongols, but it becomes too large and complicated to rule. The Tibetans, and the Mughals, and the Central Asian highlands slow the Toyotomi armies, depriving them of the conquests the empire need to survive. The empire stops growing and inevitably falls apart as empires do, and the new borders probably strongly resemble the old ones (the Japanese would probably leave local administration to the locals and have little reason to redraw lines on the map), but the culture of East Asia altered by a Japanese administration which is in place long enough to impose lasting new social structures and models of law, commerce, and political legitimacy.
Maybe there wil only be one country--China in east Asia.
Mongols, Manchurians, they once occupied whole China.However, nowadays , in Northest China(origin place of Manchurians) and Inner Mongolia (Where Chenggis Khan was born and bred up) , the majority of nationality is Han nationality and according to statistics, fewer and fewer people speak Mongol and Manchuria langualge so that government even had to establish special reserving team to help rescue that two minority language to avoid cultural loss.
And if Toyotome Hedioshi really succeeded( of course he wouldn't have considering Ming Danysty have one of world's most powerful armed force) in occupying Korea and China, perhaps Japan would follow the step of Mongols and be followed by Manchurians .Perhaps now, Japan is several provinces of China and Yamato nationality would be a minority of China and Japanese is being rescured like other minority languages.
Of course, a more powerfu country will emerge in east Asia--both with land power and sea power.
It would've been just another chapter in the long history of the ebb and flow of conquest, the changing tides of victory and the inevitable reversals of fortune that invariably follow shortly therafter. In other words, there would've no long term impact as power bids and struggles - especially in that part of the world - are short lived and impermanent at best.