How do TPE fifth-freedom rights work? I know SQ used to do SIN-TPE-LAX. But that was also before EVA set its sights on being a Star Asia-U.S. powerhouse.

SIN-USA nonstop will only work where there's a lot of premium traffic that really wants to fly nonstop. Otherwise, hubs in between will always have lower costs and better economies of scale. That's why I believe SIN-SFO will come online before SIN-YVR/SEA does. That premium traffic is probably almost all NYC, SFO, and LAX. (IAH would really be pushing it.)

How do ME3, EU3, and the Northeast Asian carriers share ASEAN-USA flows?

SQ's competitive advantage with US flights is probably going to be premium SIN-USA flows, and ASEAN-USA where existing competitors can't do as well a job. (Indonesia, maybe? But I can't see that being a big USA market for the time being.) ASEAN north of Malaysia is better flown through Northeast Asia/ME3 or even nonstop (see MNL).