Can't help with child friendliness of airlines, but two other things
regarding booking, personally I prefer booking transcontinental flights with travel agents, due to being able to talk to a person face to face rather than over the phone and I can drop the whole issue of making changes onto them if problems arise (it did help that I always interacted with the same travel agency employee so she knew how I like to do things..). Also most agencies will match prices, so they aren't more expensive, but it requires to do a little bit of research beforehand.
- if you are price sensitive, use Skyscanner, Kayak, or some deal website like that
- if you are timing sensitive, use ITAsoftware search (it's now owned by Google), it is pretty easy to enter your exact requirements and find all flights that fit. The site lists the standard airline prices (I think it includes airline specials but not those from other places) and you can't book through them, but it gives exact information the travel agent can use to build the itinerary (till now the agent never charged me more than the listed price)
Also regarding frequent flyer miles, some of the airlines are organized into groups, Singapore is in Star Alliance, Quantas (& Cathay and BA) is in Oneworld, Emirates unfortunately goes it alone... The nice thing is you can get miles for (almost) all flights of group airlines (i.e. if you have a Qantas account you get miles for flights with BA). The bad thing is that the exact rules are a major mess and headache, for example for Star alliance flights it matters what airline physically conducts the flight (for example you might have a Singapore airline ticket, but the actual flight is done by Lufthansa or Air New Zealand), for Oneworld it matters who is listed on the ticket (i.e. you might sit in a Quantas plane but your ticket is from BA....). And since Emirates and Qantas cooperate for flights to Australia, you will sit in an Emirates plane but if your ticket says Qantas you get only miles for the Qantas program (and if it says Emirates you only get Emirates points...)
My recommendation is to open an account with one program within each group (i.e. pick one in Star alliance, Oneworld, and Emirates). For Oneworld I would go with Qantas (since they have also points programs for shopping in australia and in Oneworld the rates for flying with your program airline are much better than for partners). Just make sure you open the account while still in the UK, there are some cases in Australia where they will charge you for it.
For Star Alliance I personally use a small Greek program (never flown with the airline, but it's really easy to get status), but if you only fly very occasionally have a look at the program of United airlines. Singapore's Krisflyer is known for really catering to extreme frequent flyer in business/first class...
And although I wrote that much about it, my recommendation is to pick two or three programs and when you book flights don't really use miles to decide which flight to take, it's too much effort and not enough reward (and let me warn you, collecting miles can become addictive...)