Thanks to all for the helpful and interesting replies. Here is the route we have ended up booking: Geneva-Dubai-Singapore-Auckland-Cairns-Sydney-Singapore-Colombo-Dubai-Geneva.
Here is how we booked it:
1. Return ticket on Emirates, Geneva-Singapore, coming back via Colombo (booked on Emirates website).
2. Open-jaw return ticket on Singapore Airlines, Singapore-Auckland and Sydney-Singapore (booked on SQ website)
3. One-way on Air New Zealand, Auckland-Cairns (booked on Air NZ website)
4. One-way on Qantas, Cairns-Sydney (booked on QF website)
The cost was a little more than the all-Emirates routing I mentioned in my first post, but this separate ticket option avoids some doubling back and extra domestic legs in Australia, and has generally much better departure times (Emirates seems to like taking off in the wee hours of the morning).
Some lessons I have learned in putting all this together:
1. Travel agents (at least here in Switzerland) are not any better at putting complex itineraries together at reasonable prices than a reasonably experienced traveller with a browser is. Nothing they found was cheaper or in any way better than what I found myself, and on top of that you have to pay the agent fee. Not worth it. The only advantage I can see of using an agent is that they can book and hold the various tickets until everything is organised, and then you pay in one hit. Doing it yourself on the internet, you risk booking and paying for one ticket (e.g. GVA-SIN) and then finding you can't get seats at reasonable prices on the next one (SIN-AKL), even though they were there when you checked a few minutes earlier. I tried to minimise this risk by doing the bookings as simultaneously as possible, in multiple browser windows.
2. The "multi-city" option on airline and online travel agent websites is generally useless for more than a simple stopover or open-jaw. Most of them just gave up with an error message when presented with more than 3 stops. Others would cope, but come up with absurd prices. The notable exception was the Emirates booking engine, which is very capable and flexible, produced sensible options, and only started giving silly prices if several non-Emirates sectors were included.
So, now I just have to wait for July...