Korea Government Open Data.
Open Government, Korea API is a free Government API that requires apiKey authentication, is served over HTTPS and has unconfirmed CORS support.
국가에서 보유하고 있는 다양한 데이터를『공공데이터의 제공 및 이용 활성화에 관한 법률(제11956호)』에 따라 개방하여 국민들이 보다 쉽고 용이하게 공유•활용할 수 있도록 공공데이터(Dataset)와 Open API로 제공하는 사이트입니다.
Details from data.go.kr.

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Yes. The Open Government, Korea API is listed in the Government category of the public-apis directory, which only includes APIs that are free or have a free tier. Check its documentation for rate limits and paid plans.
Yes. The Open Government, Korea API uses apiKey authentication, so you need to register for credentials before making requests.
CORS support for the Open Government, Korea API is unknown. Test a request from the browser, and fall back to a server-side call if it is blocked.
Yes. The Open Government, Korea API is served over HTTPS, so it's safe to use in production.