>> An American Airlines rep told me that Qantas first makes award seats available to their own FF'ers before making award seats available to American FF'ers.<<
Which is true, but with an asterisk. Qantas opens its booking windows a month earlier than AA does, so Qantas FFers get 360 days lead time rather than 330 for AA FFers wanting to redeem miles on QF flights. So it's a timing issue, not one of priorities in the queue.
But I have never experienced a case where a Qantas seat shown as available on the QF site DURING THE AA OPEN WINDOW wasn't available for redemption using AA miles. So unless you're looking 12 months out instead of 11, and you see something that will work, then it ought to work using AA miles.
Seats are released into mileage redemption inventory on a rolling basis - you might not see anything available at T-11 months, but seats may well appear right up until a few days before flight time, depending on how Qantas' (or any airline's) yield management computers make their real-time judgements. I just booked a mileage seat on a rather hard-to-get route for a flight next week; looking out several months there's nothing, but closer in, there's space.