Upvote:6
Sorry for your troubles but most of this unfortunately your own fault.
The FAA clearly states that any person past their second birthday must be in their own seat: http://www.airweb.faa.gov/Regulatory_and_Guidance_Library/rgFAR.nsf/0/2CE6D9B34E1C56578625708A00720A50?OpenDocument
Emirates could not allow you to board without a seat for your child, so they did what they had to.
When you booked with Qantas you agreed to the rules and conditions of the booking, which in all likelihood would clearly state that your child needs to be below 2 years of age to qualify as a lap child. You agreed to this.
First and foremost, it's your job to make sure you are complying with all required rules and regulations (such as visa, entry requirement, etc). You knew that your child would turn two before the return flight, but you didn't check with the airlines or did a quick Google search which would have showed immediately that this can't work.
I'm afraid you don't have many options here other then "painful lesson learned".