We went to Oz almost 2 years ago with our 3 who were 4, 2.5 and 13mo.
We flew straight through both journeys (stopped for an hour and a half for refuel in Singapore) and I am glad we did it that way to be honest. It felt like we managed to get the flights out of the way and done with, I think it would have been awful doing a 13 hour flight and stopping off for a few days knowing we still had another 8 hours to go before we got to Oz.
The kids coped absolutely fine on the flight. The eldest 2 had packs from the staff (Qantas) which didn't keep them occupied for very long to be honest but it was at least something for them to open and inspect! We had taken loads of toys for them but they were far more interested in their personal TV's with hundreds of movies, programmes and games to keep them occupied. They slept really well, we took their own little blankets and their bedtime bear they each had at the time and snuggled them up on their seat with a pillow. The youngest was in a bassinette. They didn't really have any jetlag. Our flight was at midday from Heathrow and we arrived in Sydney at around 6pm iirc. By the time we got to the apartment the kids went to bed by about 9pm but then all woke up at 2 in the morning wide awake! Took us a few hours to get them back to sleep and then they slept til about 11am that morning. We were up and raring to go before they were! We then just made sure they had a really busy day, didn't let the youngest nap at all and they went to bed about 8:30 that night and were back into their normal routines the rest of the holiday.
We went for just under a month and spent 4 days in Sydney then flew to Hamilton Island, stayed with my DBro and family for a short while and then very slowly drove back from there to Sydney (having a holiday on the Sunshine Coast with my DB on the way, who then drove back up north while we carried on down south). Then had a few more days in Sydney before flying home.
Unless you have a huge desire to visit Singapore, I would say don't bother with the 3 day stopover. 2 weeks really isn't long enough to incorporate such a long stopover, I don't think so anyway. As other have said, there is plenty to do in Sydney and surrounding areas, I would stay there as long as you can or look at extending your trip if possible.
Have fun, I'm sure it will be great x