arun, here is
fewmiles' website on Platinum & Gold Challenge - the
undocumented feature of AAdvantage (that's why you couldn't find it). They aren't about to shout it from the rooftops.
AA Mileage Page
QF Mileage Page
As you can see, the AA flights will earn 100% EQP's. The LAX/JFK/LAX AA flights might be enough for a Gold Challenge. According to the QF website, the distance LAX/JFK is 2,467 miles; so you are a whisker short of a Gold Challenge.
In the QF Mileage chart, you'll see that Q doesn't show anymore. Not sure whether it's purposely missing (ie. nil mileage, nil EQP's) or by mistake.
AmericaWest isn't a valid airline (unless there is an AA codeshare), so no help there.
When you join your wife and son to AA online, you'll be able to print out temporary cards. Real cards don't get printed until you have flown sectors that have credited to the account. But I printed out my cards and just made sure the AA number was in the booking, checked with the CSA upon checking in and again getting BP's (where they should show again).
As I said to
Bob, don't ring AA SYD to link everybody to the AA Gold or Plat Challenge, as they have not been well trained about this. Take a minute longer to do it right the first time, by ringing the number that I gave
Bob in the link he has shown (look at my first reply).
If the QF flights don't get you credit on the AA program, then look to give your QF number for them at check-in.
AAdvantage Gold (oneworld Ruby) benefits:
Priority check-in at Business Class desks (where available), regardless of the class of service flown that day
Preferred seating (where pre-reserved seats are offered)
Priority standby and wait listing