According to the details above, I'm going to assume that the PoD occurred in 1449 (Tumu Crisis), while the map is portraying the situation around the 17-18th century, given that you're mentioning Europeans within India. If these assumptions are correct, the (Later) Yuan would almost certainly not exist after 100-200 years or so, given that a group of people are eventually going to justify the loss of the Mandate of Heaven as revolts erupt across the country, most likely due to severe famines. In addition, once this dynasty begins to crumble, the Cai and Korea (using "Jin" for the "new" dynasty is questionable, which I will discuss further below) will seek to recapture their "lost" territories, causing the nomads to be pushed back into the steppes. In any case, even if the Yuan somehow manages to hold onto North China for longer than two centuries, which in itself would be virtually impossible without severe social upheaval, the ruling class would have been assimilated into Chinese culture within several generations, given what happened to the (Jurchen) Jin, Yuan, and Qing IOTL, causing control to eventually revert to the Han Chinese majority after social disruptions eventually contribute to the end of the dynasty.
Korea's portrayal is also questionable for two main reasons, specifically concerning "Jin" and the Yuan "conquest" of Korea's northern territories. "Jin" doesn't exactly make sense in context, as "Goryeo" and "Joseon" were named in reference to prior well-known dynasties ("Go(gu)ryeo" and "(Go)Joseon," respectively), not to mention that no Korean dynasty was referred to with just one syllable, barring a questionable exception, so you'll have to be specific on exactly which Chinese character "Jin" refers to in this scenario.
The territory that you're depicting is also questionable because for close to 700 years, the Khitan Liao, Jurchen Jin, Yuan, and Qing all failed to "conquer" and hold onto the peninsula's northern regions, as they essentially withdrew multiple times when the Korean court agreed to pay tribute. In particular, Goryeo held out against the Yuan for over 40 years precisely because the military faction had been in control, which will not be the case here considering that Sejo would probably take the throne around 1450-3 due to butterflies. In other words, while the court would probably be holed up in Ganghwa Island, the Mongols would almost certainly refrain from continuously attacking the peninsula for more than 10-20 years or so given their prior experiences, as the Mongols have less resources to spare due to controlling much less territory, not to mention that China is still holding out. Although the Ssangyeong and Dongnyeong Prefectures were incorporated as part of the Yuan, they depict a different scenario because they were annexed during peacetime, not to mention that the former was eventually returned after 20 years as a result of the Goryeo court continuing to stubbornly petition the Yuan for their return.
Joseon would also be well-prepared for a foreign invasion, as although the peninsula was caught entirely unprepared during the Imjin War, well over 150,000 Korean soldiers participated, not to mention that the navy led by Yi Sun-shin continued to disrupt Japanese supplies. In addition, Sejong ordered his generals to invade Tsushima with around 18,000 troops in 1419, and sent 15,000 against the Jurchen in 1433, both consisting of minor expeditions suggesting that at least 50,000 troops had been stationed across the country around 1400-1450, which could easily have been increased to around 100,000-200,000 in the event of a major war. It would also be extremely unlikely for a general to stage a coup, given that he would be warmongering, which would face extreme resistance from the people weakened after years of war. On the other hand, once the (Later) Yuan begins to show signs of strain, Korea might begin to push further north into Liaodong and Southern Manchuria, given that Goryeo refugees in Manchuria IOTL caused the Yuan to assign Shenyang (Simyang), stretching from Liaodong to the Tumen River, to Goryeo princes, and was a major reason for Goryeo's ambitions in Manchuria shortly before its fall. It's also possible that when the Mongols conquer the Jin remnants, Joseon might ally with the Yuan to expand further north, increasing the likelihood of Koreans within the region.
While this might work for Vietnam, I'm not sure where you got the impression that Korean dynasties can be referred to solely by their family names, as some dynasties were ruled by more than one clan due to political infighting, while some shared the same surname. According to this logic, the "Wang" dynasty was succeeded by the "Yi" dynasty, not to mention the "Hae," "Hae/Go," "Buyeo" (different from the state of Buyeo which officially ceased to exist in AD 494), "Kim," and "Park/Seok/Kim" dynasties that existed beforehand. Joseon is a relative exception because it can also be specified as the "Joseon Yi," but "Yi (Lee)" is never used by itself to solely refer to the dynasty in question.
In addition, titles of Korean dynasties had more than one syllable with only one exception (Jin), while the vast majority of Korean surnames only had one syllable ("Buyeo" was the only exception for a royal family), so that should have also suggested that the names of specific dynasties were unrelated to the royal families' surnames in question.
Agreed here.