You can certainly earn AA Miles on the DFW-LAX and AsiaMiles on the LAX-HKG-BKK legs. If the Agent doesn't know how to enter it then either try again until you get a more competent agent or wait until you check-in and sort it out then. The important thing is that when you get on the flight the BP must show the FF account that you want your miles to go to.

The only thing you can't do is earn AAdvantage miles on the CX flights, because M-class isn't eligible.