Not rude, but holding eye contact (=staring) is uncomfortable for the listener. You could be mistaken for flirting, as well.

Basically, eye contact is also an important thing during communication in Korea. But if you talk with elderly people, making eye contact might be considered as a rude attitude.

nope

Hello, Mexichunt. Not really. In Korea, however, avoiding one's eyes while talking may not mean 'I am not listening to you'. Especially when they talk with someone older or socially standing higher, they sometimes drop their eyes as the submissive response.