Am I capable of F-4 visa?

First, I am a woman, and I don't need to go to military in order to live in Korea.

So currently I am a citizen of Argentina and USA. Throughout searching and asking in order to live in korea, I found out I also have korea natinality; they were just not listed under my parent's family tree registry. After speaking with Korea consular, she said first I need to be under my parents family tree registry and then give up the korean nationality. For I know I just need to give up one of the three nationality I have(USA, Argentina, and Korea).

1. Is it really necessary for me to give up korean nationality. Can I not give up Argentina nationality instead and keep USA and Korea nationality?

2. Can I still apply for F-4 visa even though I have korea nationality?

3. Do you know how long it takes to be under family tree registry?