I'm not sure exactly how child and infant tickets come into play. Presumably you'll need to purchase an adult ticket, a child ticket (at the same cost as an adult ticket) and an infant ticket for which you only need to taxes. Though I may be wrong on that.
For BNE-CNS in J:
QF will charge 24K points per adult plus taxes and fuel surcharge
VA (Velocity) will charge 21.8K points plus taxes
AA (for travel on Qantas) charge 17.5K points plus taxes
No idea about Elevate, sorry.
AA generally offers the best value for redemptions with Australia and NZ as they are treated as one "zone" and any redemptions within that zone are the same price (be it Sydney-Canberra or Perth-Auckland). By comparison, VA and QF price their awards according to distance so for shorter flights they come out on top (just), assuming points have the same value across all programs.