Z is correct for First Class reward on QF, U is business class and for domestic business class (same as USA first). I have done this several times, and you are OK.

Since Z is the same code letter as a BF ticket on CO, when CO prints out the paper ticket - the itineray card (first card) will type out as Business Class since CO computers are picking up the title from the Z class look up.

Rest assured that you will be in First for Z class. The key is the code on the actual flight coupon - the class should show Z class for LAX to SYD (for example) and the status should be OK. Australia domestic flight coupons will show U for QF.

Your CO flights should show "E" if you are in CO FC, and "M" if you are in CO domestic coach. Might check those to make sure CO put you in FC for the domestic flights. If it shows M, be prepared to fly CO coach - while they say you will be waitlisted - it will never clear. I have done this 10 times.

Also, call QF and get your itinerary number and do your seat assignment. You also need to get an Australia electronic VISA. You used to be able to call QF to get the VISA, but now they charge you for it. Best to call CO desk and have them get the VISA - last time CO did not charge for that service.