Just turn up for the flight with the required ID and you will be fine!
True, but that's not much use if he wants to check in online, or book meals...
It also brings a problem that typically immigration ask you to carry a copy of your itinerary as proof you will leave the country. Not usually checked, but if they ask for it I don't think 'well, the airline didn't email me it and I lost the booking reference so I couldn't retrieve it myself and when I rang up CS they wouldn't help me...' will suffice.
As TightSlot asks - has the payment been processed? If it hasn't, then the booking won't be in the system.
Why Qantas CS won't help, given the verification of a few facts I don't know. Perhaps asking to speak to a supervisor?