Sydney to Washington: Qantas or American Airlines?

No brainer QF A380 to DFW...the long flight is MUCH more pleasurable in the A380 as its super quiet, high humidity levels and lower cabin pressure altitude setting. Food and service is much better too. Only tolerate AA(or any other us carrier) as long as you absolutely must.

I assume you're only looking at QF for FF rewards, fair enough. Admittedly, I'm a UA FF, however, the UA Dreamliner across the Pacific is fantastic, particularly in J with the new improvements. You could go via LAX or SFO - well connected as LAX, SFO and IAD are all UA hubs.

Out of Syd or MEL yes QF7 over (after short MEL-Syd of course if needed), but if coming back I recommend QF94 - the tBIT connector really helps on way back, the lounges are better, and you don't need the extra hop after the long flight.

Yes the QF7 option through Dallas is the better choice. On the connection at Dallas bear in mind that the AA flights go from a terminal on the other side of the airport campus, connected by an elevated train, so this is not a quick transfer for pasengers and luggage will need time too. Also the food outlets in the arriving terminal are probably better than in the departing terminal - it is certainly less crowded.
For Washington Regan is so much closer to downtown than Dulles and in our experience works quite well.