Joseon Dinasty period in Korea. Neo-Confucianism on the rise, so shifting focus towards more rationalist view of the world and away from superstitions and mystical elements of Buddhism and Taoism

This is not how I would characterize what Neo-Confucianism and Buddhism are/were and the effect they had on Korean society, and why Neo-Confucianism was adopted over Buddhism. Neo-Confucianism, though this is also a simplification, had just as many mystical and spiritualist elements available to its practitioners as Korean Buddhism or Korean Shamanism did. Indeed, the three weren't necessarily mutually exclusive.

Buddhism is a very logical religion. I don't mean it in the esoteric sense, I mean it in the "Buddhist monks spend a lot of time on logic and epistemology" sense. Korean Buddhism is no exception. Korean Buddhists are quite proud of the use of movable type printing presses that were used to print the Buddhist Canon, the "greatest" edition of which is housed in Haeinsa, one of Korea's most prominent and prestigious Buddhist temples.

On the other end of the spectrum, Confucianism has historically never shied away from mystical or metaphysical elements of religion. One of the primary books of the Confucian canon is the Yijing, which is often considered a book on moral reality and judgment, but also serves as a model of the universe, a treatise on the nature of heaven and earth, and a divination manual. Famously, Admiral Yi Sun-shin consulted the Yijing throughout the war, and recorded some of the findings (as well as his dreams) in his War Diary. Ironically, the host of the history of Korea podcast actually used this detail to come to the reverse conclusion, that Neo-Confucianism was more shamanic and mystical than its predecessor. An equally mistaken conclusion.

Anyway, (Neo-)Confucianism and Buddhism have both logical and mystical elements. Korean Buddhist temples, while they do spend a lot of time in logic and epistemology for monks dedicated to textual study, also house the "Mountain Spirit" (san shin) in its own shrine (almost universally in basically every Korean temple I've been to). Meanwhile the Confucian rule of Joseon led to the "Hall of Worthies" and a series of pre-scientific advances such as the rain gauge, a proportional tax system, advances in naval technology, the development of a Korean alphabet (Hunminjeongum), among others.

What the changeover from Buddhism to Neo-Confucianism in represented was more of a change in administration and administrative structure than one of philosophical difference. The Buddhist temples in Goryeo Korea ended up quite corrupt. Though whether this is actually true is not my particular area of expertise. At the very least, they were a drain on resources that the new Joseon government wanted to tap into, and sought a thorough change of governance. Since the Mongols were also very close to the Buddhists, and the primary changeover from the Goryeo to the Joseon was based around divided loyalties between the Goryeo (who supported the Mongols) and the Joseon (who supported the Ming Chinese).

The Muromachi period (Ashikaga) in Japan. Japan is very militarized. (As a response to what's happening on mainland?)

I suppose it depends on what you mean by "militarized." I'd say it's also a vast generalization of a period that lasted two and a half centuries (for comparison, would you call the period of American Independence (1789-Present) one of "militarization"?).

It was certainly a period of challenge and division, and a breakdown of traditional order. Not universally, but mid-way through we get the Onin War, which is still known as one of the traumatic experiences in Japanese history until the 20th Century. The resultant loss of central control led to the Sengoku Jidai, when neither the Emperor nor the Shogun was able to exercise any central authority in the country, leading to the daimyo (lords) doing basically whatever they wanted. All until Oda Nobunaga set out more-or-less conquering his way to being basically Shogun.

Yuan dynasty in Mongolia. Ming dynasty in China. This area seems a little messy and confusing to me.

Temujin was elected Khaan in in the Khuriltai of 1206. He and the Mongols launched and invasion of the Jurchen-led Jin Dynasty of modern north-east China in 1211. In a later campaign the Tangut-led Xixia Dynasty of modern Qinghai Province rebelled against the Mongols, choosing not to offer troops in a campaign and Temujin, now known to us as Chinggis Khaan (literally meaning "King of the World") set out on a campaign of retribution. It would turn out to be his last campaign and he died in 1227.

Temujin's grandson, known to us as Khubilai Khaan, completed their comquest of Song Dynasty China in 1279. This was after his first attempt at conquering Japan (1275) aided largely by Korean shipwrights. A second attempt was launched in 1281 with equally disastrous results. The Mongol conquest of Song ended with Khubilai Khaan announcing the creation of the Yuan Dynasty.

The Yuan would end up retreating back to Mongolia in 1368 as the Hongwu Emperor came into power and announced the creation of the Ming Dynasty in China. The Mongols would claim to be the true rulers of China at least until Mandukhai Khaatun and Dayan Khaan (Dayan was a name chosen specifically to strike fear into the Ming, sounding like "Da yuan" (i.e. Great Yuan) to Chinese ears) managed to reach an accord with the Ming in the 15th century, and at the latest in the 18th Century when the Mongols elected the Manchu Emperor in a new khurilatai as the new Khagan of Mongolia.

What is confusing me is, does that mean that there was no real Mongolia in the shape that we know it now, but it was just spread out into China?

I'm not totally sure what you mean by "real Mongolia." There very much was a complex society in Mongolia during the time of the Yuan Dynasty (and before and after it, too). Khubilai Khaan came to power ultimately after a sort of "Mongol Civil War." The War was between Khubilai and his brother Ariq Boke. They were the sons of Tolui, Temujin's youngest son born to Borte Khaatun, and in the traditional Mongol fashion, the oldest sons were sent abroad to expand the borders of the tribe, while the youngest son was kept close to home to inherit the house of his parents. By that time things had become more complex as rivalries between Jochi and Ogedei, the oldest sons, became a driving factor in Mongol politics.

Either way, Mongolia was very much a power player in the time of the Yuan Dynasty. Ariq Boke consiered himself Khaan of Mongolia while his brother Khubilai was the Khaan of China. Khubilai disagreed, thinking that he inherited their grandfather's position as King of the entire World, not just China, and that Ariq Boke owed him fealty regardless of what corner of the world was his. In this case, Mongolia, which was home to some of the holiest Mongol sites, and had a complex and somewhat advanced city that was briefly the capital of the entire Khaganate (Karakorum).

There's a lot of different sources for these, too many at this point. But if you're interested in a specific aspect of things I've written above I can point you in the right direction. There's also the r/Askhistorians booklist which has some good details on Korea and China. The one that occurs to me that would be the most general about Korea is Korea: Old and New edited by Eckert and Lee.

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