can log shipping to log shipping? I have one production SQL Server 2008 Database with log shipping set up for that server in US.

I have a account in the log shipping database so I could sometimes run query on log shipping database.

Now I need to ship data back to Korea daily but I don't want to impact production performance. Can you suggest any technique to achieve this
You can ship the logs to as many sources as you want. Just add a new secondary server on the primary one. The impact will be on the Korean server as it will have to copy the logs from the backup directory and replay them.