Yes, you can.
If you provide the QF number but are not creditable class to QF such as Q,S class. (Lowest would be V eligible for QF credit)
You can retro claim the miles from Asiamiles.
Most likely the TG sector is on UA award. Qantas no longer interlines on separate tickets.
So you definitely need to collect luggage from domestic and go to International and check it in yourself.
As regards to MH leg, you could probably ask TG to tag it through all the way from SYD to BKK.