I used to assume "Style" was English and couldn't be Korean, but I
subsequently found out that there's a Korean word based on the English
It's true that Korean has adopted the word 'style' into everyday speech, so a Korean might not feel they're "speaking English" when they use it, but even though it is a Korean word, you could still call it an English word too - it's spoken basically the same and means the same thing. https://en.wikipedia.org/wiki/Loanword goes into this concept. But by your definition, it would be in a Korean dictionary.
Anyway, back to the song..
In that page you can also spot
"Eh, sexy lady" (I always heard it more as Hey, sexy lady)
"You know what I'm saying"
which are all English.
I don't think 'Korea' is mentioned - your ear may be catching 'geu wie' in the line Ttwineun nom geu wie naneun nom, which means something like 'the running man has the flying man above him'. (I think this is a saying that means "there's always room for improvement", "always room at the top... " - something like that.)