As far as I know, Domestic would apply to US operations of scheduled routes.
Flag Carriers would be scheduled international operations that originate in the US or a territory and terminate outside of the US. Those airlines typically fall under the rules of the state in which they were registered. So a US airline flying to the UK would be a US Flag carrier. Some operations to Canada are exempted and treated like a domestic flight.
But it would also apply to an airline operated by the State under the national flag of that country. I don't think we really have many official Flag Carriers in the US (other than standard international operations), but there are a few overseas, or there used to be. Ex: British Airways was the UK Flag Carrier until it became a private company. Same with KLM being the Dutch Flag Carrier.
Supplemental Operations are usually something like a flight test or repositioning, somethingfor-hire operation that doesn't fall under the normal operations of the 121 carrier and are generally unscheduled operations. This would include a passenger charter of an airliner.
FAR 119.3 has the definitions as they apply to US carriers. http://www.gpo.gov/fdsys/pkg/CFR-2011-title14-vol3/pdf/CFR-2011-title14-vol3-sec119-3.pdf