The BA check in desks don't open until about 1pm (that was before they moved the flights earlier). I have seen them open just after midday occasionally.
As CWS said, if you're flying in on QF code, you should be able to interline your bags, and I highly recommend doing this so (with your ownward BP) you can get the QF bus from the domestic to the international terminal airside without having to go landside and pay $6 on the train with all the schlepping to the bowls of the airport or about $4-something on the bus (ridicolous prices but that's another story).
Don't worry about the SIN connection, extremely common and it's one of the easiest places to do a transist. You will have to reclear security which is right at the gate anyway.