It depends how good you are at sleeping in economy. I can't.
When flying economy, I prefer to land on the evening and go to bed straight.
Not so much for jet lag but for lack of sleep, especially if the flight departed in the evening, live the BA from London at 10pm
I have been awake since 6 am (so 16 or so hours before flight) then the 20 something hours, means, I haven't slept for 36-38 hours, and I am dead on my feet not for the jet lag, but because of sleep depravation.
When flying business, I don't care, I have my flat bed.
For jetlag, there are products which are meant to alter circadian cycles, like the FlyKitt.
Accomodation around Christmas will be tricky so yes November is a better choice. AirBnb is something to look at. As others have said, you will be shocked by the price of eating and drinking.
Airlines wise, BA is always ALWAYS packed no matter the time of year. Singapore, Etihad, Emirates, Cathay, ... all fine. If going through Hong Kong, there is a second bag check right before the plane doors and all bottles are taken, so don't buy a drink for the flight.
Instead of buying multiple plug converters, just buy 1 and take a multi socket, so you can charge several things with just one. Look for an universal one, so no matter the country you are having a stopover , you can charge phone, iPad, ...
We are always short on time, so we never break the trip, even if for me, it is 3 flights to go home (Monaco) and the trip gets close to 27 hours. Consider that you lose days travelling. If you leave UK on the Wednesday night, you land on the Friday morning in Sydney. On the way back, it is just one day, say you leave Friday, you land Saturday.
Take a book or download movies on iPad, because once, my DD's tv was broken, so we swapped seat, and it was the BA12 so the same aircraft and same seat after refuelling in Singapore.
Yes to travel insurance. Medicare covers reciprocal health, but ambulance is not included, would you need one.
Australia is very different for heat. Even on the hottest day, temperature will go down very quickly once it is dark. And one day can be 40, the next 12.
November is a great time of the year. However, the water will still be freezing!!!! Sydney is a gorgeous city, You will love it. Make sure to come my way, the Northern Beaches. Very little to no tourists in Freshwater,