Given the current presidential scandal in South Korea, I am very interested in the mechanisms of their government. It seems that there is no specific criteria for being a minister. In my country, ministers are usually also lawmakers, but in South Korea, the president seem to pick anyone she likes. Someone care to debunk?
The Prime Minister of South Korea is appointed by the President and should be confirmed by the majority in the National Assembly. As you mentioned, there is no specific criteria for nominating a candidate for Prime Minister, except that a nominee should not be serving in the military at the time of nomination. In other words, only civilians can be nominated for Prime Minister in Korea.
As confirmation requires a majority vote in the National Assembly, usually former or current lawmakers or ministers are nominated.
There have been many cases where nominees were forced to drop out because of their unethical behaviors and politics in history. Three candidates couldn't make through the National Assembly hearing process under the current president for the last four years because of (1) his speculative investment in real estate using secret information, (2) earning too much money when working as a lawyer, and (3) his historical viewpoint that lawmakers and general public don't endorse or agree to.
Yes, of course, the President can nominate whoever (s)he wants, but only a few can go through the confirmation hearing and become Prime Minister.
The current Prime Minister is ex-Minister of Ministry of Justice (equivalent of US Attorney General). For more information, you can read the Wikipedia article on Prime Minister of South Korea.
All other ministers are appointed by the President at the recommendation of the current Prime Minister. They don't need the approval from the National Assembly, but they go through the same vigorous hearing process and they can also be disqualified and the President can withdraw the nomination in the worst case scenario.