Each car has a fixed (in the parcel shelf, roof, or boot of the car) anchor for the extra belt. The anchor belt is attached to the actual car seat, so there is no point bringing your UK one to Oz. The tether belts don't come off the seat.
You can hire car seats from the red cross here (Adelaide) so there maybe places where you can do the same there. Babies aren't allowed to travel in the front of the car here, either.
How long are you coming for? I would recommend bring nappies for the trip & asking your hosts to buy you some form here. They take up valuable space in suitcases!
Call the airline & book the bassinet. But be aware that if you hit turbulence, you have to take baby out. We hit quite a lot & it was a pain, as each time she settled, we had to lift her!
I am pretty sure Singapore did strollers on loan. We booked ours in. in the UK & didn't see it until Oz. They don't tend to unload during 'transiting' even hold baggage.
And if you are going to be there a good few hours, think about booking into the transit hotel. That was a god send for us!
Definitely change of clothes for all! Practice changing nappies in confided spaces! 
Also, small thing...expect your baby to be cooed over...a lot! I had people asking for pictures with dd. She was blonde at time & blue eyed & it was a combination that the Singaporeans (is that a word) loved! I mention it, because it greatly unsettles some people!