@heartsings unless you own assets in South Korea or intend living there, no downside becoming a US citizen. You can still freely visit home country for 90 days visa free after becoming American.
In terms of documentation, everything @Crazy Cat suggested should be good.
Also, if you pay child support, you need proof of that. If you had any criminal convictions or citations, you need to have court docs for it. But in that case, make sure to consult with a lawyer first.
