Shipping from USA to Japan

I am going to be shipping my shamisen to Japan to its maker for repair and to have karitsugu (wood joint guards) made for my instrument.

I just got my shamisen all packed up with bubble wrap in the mitsuori case and I am going to pad it inside another box for shock absorption.

But I don’t know what the process of shipping something like an expensive instrument is like…

Are there customs export laws against shipping kouki wood to Japan?
What shipping service (usps, ups, fedex?) is best to use for a safe arrival?
What is the general process/steps of doing so?