Well done on the Green Card! I'm a QF WP who has moved to New York, so I do the JFK-LAX-BNE run pretty often (always in Y), so here are some tips and observations that I found:
- You won't have any hassle with the third bag. I do it all the time and it's pretty normal for the check-in staff. I also have a wife and a 3-yo son who are QF PS, so they have the same allowance. In truth, we've never actually hit 9 bags, but did 7 last trip with no problems. Make sure you're under 32.0 kg. They can't/won't bend on this.
- Luggage transfer can/will be tricky, depending purely on luck. You'll arrive in Tom Bradley Terminal in LAX where you'll have to get through Customs; trolleys are free and easily available after passport control. You'll need to recheck your luggage for QF107 LAX-JFK. On a good day, you can do this in the same terminal, but it doesn't always work that way. If the recheck area's closed, you'll need to physically get your luggage over to Terminal 4 (you can take your trolley). You won't need any bus. It's a 5 minute walk. This may help you get a feel for how much luggage you can physically carry; at LAX, you'll get help from nobody. Don't forget if you have big carry-on, you'll have to lug that too.
- Because you'll be out of security, you'll need to clear security again. There's an express queue in Terminal 4 which should whisk you through pretty quickly. It's supposed to be for AA premium members only, but it's worth a try if you have Qantas Club on your boarding pass. They're often not smart enough to figure out the difference, so that should save you a bunch of time (at least 2 hours at peak times), so it's worth the effort.
- The QP in LAX is a shared lounge with AA, and it's not very good. Make your way to the upstairs desk once you're in straight away if you want a shower because they fill up quickly once the AUS flights arrive.
- In JFK, you'll obviously need a trolley again. It will cost $3 which can either be dollar notes or Amex. They do priority tagged luggage pretty well there, so it shouldn't be a long wait.
- Once you're outside, get a cab into town. I assume you're going to Manhattan, becuase if you are, it's a flat fare (about $45) PLUS tolls and tips. For the other boroughs in NYC, it's the metered fare, again, plus tip. If you're going to Staten Island, you'll need to cough up for the toll on the Verezzano Bridge too ($10). They can (and will) charge extra (about $1) for each bag, but putting them in the seat if the boot's full won't be a drama.
I hope this helps and if you need any help in the Big Apple, give me a shout.
On my own, it is going to look slightly ridiculous! 
