Questions about example sentences with, and the definition and usage of "Ministry"

A: Minister is the person responsible for (education).

A: “According to the National Life Survey of the Ministry of Health, Labor and Welfare, although Japan is the world's third largest economy after the United States and China, 1 child in 7 under the age of 18 lives in poverty.”

A: >> Ministry

A: That sounds correct.

A: Your writing is already very natural-sounding, so it is easy to check! ^^

A: × The Ministry of Education disliked this argument and even issued a notice in 1963 stating that it was not appropriate to admit students considered unable to complete the high school curriculum.
✓ The Ministry of Education disagreed, and even issued a notice in 1963 stating that it was not appropriate to admit students considered unable to complete the high school curriculum.

× In response, some high schools had shown their resistance to this; they began to admit all students, even if they exceeded their capacity, but these schools often failed because they also attracted students with poor grades and behavior who could not be admitted to other schools.
✓ In response, some high schools showed resistance to this; they began to admit all students, even if they exceeded their capacity, but these schools often failed because they also attracted students with poor grades and behavior who could not be admitted to other schools.

"disliked this argument" is ok too
I would just say "Some high schools resisted this;"

A: × According to the Ministry of Foreign Affairs, 64 Vietnamese trainees aiming to become nurses and care workers will arrive in Japan on Monday, and other 167 will arrive later this week, about six months delay due to the pandemic of the novel coronavirus.
✓ According to the Ministry of Foreign Affairs, 64 Vietnamese trainees aiming to become nurses and care workers will arrive in Japan on Monday, and another 167 will arrive later this week, about six months late due to the pandemic of the novel coronavirus.

A: Yes, but you don’t have to capitalize anything except for Japanese. You only have to capitalize things when it’s the beginning of a sentence, an name or a person or “group” such as a race, or a special place. :)

A: "...I am all for making a big profit, but complicated factors, including an aging population, westernized eating habits, fewer farmers, and a low birth rate have all contributed to Japan's lack of food self-sufficiency."
"I think it would be better for Japan to buy more food from different countries so that we could secure a stable food supply rather than stick to this ratio."

Overall, your English is very good! There were a few phrases that were a bit redundant, along with some small grammar issues, but it was very well-constructed!