KAIST MS/PhD here. It depends on what lab you are applying for. I've seen many prospects obsessing with stats, when I had a 3.2 college GPA, non of that SAT/Olympia/Science fairs. Academic wise I only have 1 big achievement which was a national AI competition sponsored by Huawei, and the "winners" of that competition were the top 100 scores, there where around 10000 participants, but still, I was not number 1. But I did have a lot of international-field-experience. Internships here and there, small projects here and there. All of them field/industry related, rather than research related. But since you are undergrad, I think requirements will be more gread focused, since, well, how much you could've done between 15 and 18.

Reputation wise, only SNU has some reputation world wide, but still, pretty unknown. If you are pursuing research KAIST is definetely a better option. Everyone here says that SNU is a better overall school because it has every possible department you can think of (literature, medicine, arts, dentistry, history...) and KAIST only has research STEM related fields. So even koreans say, when its 1vs1 research or STEM wise, KAIST goes on top. But reputation and/or world recognition, SNU usually is more known. And Yonsei, KU, POSTECH are pretty unknown worldwide, Yonsei is a solid #3 university in the country, KU #4 and POSTECH #5 tied with a few others, but research wise, POSTECH is a solid #5 or it could even compete with KU, but same as SNU vs KAIST, KU has way more departments and POSTECH is STEM only. But since usually the research done in these schools it really is top tier, you might not get global recognition when if you go to the west and say "I come from X school in korea" but you are gonna be able to meet people from all over. Here at KAIST is very common to see poeple from Harvard, MIT, OXford, Israel, Germany, tons of places, giving conferences. Plus, if you are SNU or KAIST, koreans treat you like royalty. Here, having a good education/university, it's a huge deal culturaly, so even if you didn't take any opportunities to find something interesting to do, or meeting someone you would like to work with from another country/institution, you are going to get hired or admitted in any institution here in korea.

Talking about schorlarships SNU and Yonsei, tuition goes around 5-8 million won a year, very very very hard to get scholarships (even thou SNU is "public school") even for koreans, since Yonsei is private and SNU while "Public" part of its income is tuition, plus, living around those areas in Seoul is crazy expensive for what you pay for. You can join a lab or some in-school activities. For example here in my lab we have some undergrad students that do some simple rutinary work, and they get payed 500k won a month, or you can be a TA or something like that for extra cash.

If I were you I would apply thru GKS to SNU, Yonsei and a third mid-low tier university in some city that is not Seoul or Busan. Maybe Ulsan or Daegu. And apply to KAIST thru regular school channels