There is no such carrier as "China Air". There's Air China (CA), a *A member. There's then China Airlines (CI), a Taiwanese carrier that recently joined SkyTeam. China Eastern (MU) and China Southern (CZ) are also SkyTeam members. Vietnam Airlines (VN) is also an option within SE Asia.

I agree with Paul3456 that you should aim for KE. The 77W out of ORD seems to show good award inventory, and the flat bed seats are great. A380 out of JFK can have good inventory, too, but be aware that they're pulling one of the A380 JFK flights soon to start running it to FRA.

You can't book international F with SkyMiles, but if you need domestic Chinese flights, I've seen reports of people booking the CZ and MU F product on two-cabin flights there like you can book domestic F in the US. Since you're going to BKK, that's likely not going to be something you need unless you run up against the KE blackout dates.