Do I have to pick up my luggage during the transit at Xiamen Aiport to Singapore?

Oh, yes, you have to pick up your luggage and reclaim it if you will transfer from international to international. The procedure is like this:
Flight Arriving - Arrival Access - Inspection & Quarntine — Immigration— Luggage Claim—
Customs — Internation Arrival Hall — Go To 2F for Boarding.

What if it is domestic to international? Same procedure?

You will follow different procedures to transfer from domestic to international. Here are the procedures: Flight Arrival- Luggage Claim- Quarantine & Inspection- Customs Inspection- Check in- Immigration Inspection- Security Check- Waiting and Boarding. There are clear English guiding signs for your convenience. Take it easy! :)