In most airline alliance bookings, (I have done it when Lan was still in OW a few years back, put in my QFF # number pre flight, this was with a LA flight, so earnt less QFF than flying its own plane, and less than when the LA flight had its QF code), when you make airline member bookings in OW, if on the AY Finnair website for eg, it will ask you to pick from the drop down menu, your OW member airline name ie QFF and also will also ask you to enter your member number.
Not sure about Skyscanner, and can't say about that one.
OT, I have made bookings with Flybuys Travel Aust, for VA flights, and it does ask for your FF number there too, and also asks for FB # number when you log in.
VA is not in a major alliance, though this won't affect you.
If you book with the AY website, and have provided your QFF number, as I mentioned, a lot of the time, things go to plan, and the your QFF member number will appear on the BP and, and down the track, your QFF account.
Sometimes, it does not, so always keep BP and PNR details for a few weeks after the trip.
If you have QFF SG which gives lounge access, you should bring along that card, but as you are flying disc J, you will get lounge access at most stages.
In SYD, as BA does not have its own lounge for eg, you will get access to the SYD QFi J lounge, if its open by mid this year, or the very good QFi F lounge, which has a better view, plus better food too.
In SIN, not sure, if you get BA's lounge, or the you might get the SATS dnata lounge, as I think AY does not have a lounge there.
With Covid, the lounges overseas might be closed, but for SYD, most likely it would be the better QFi F lounge, (my paws are crossed for you.
I have used both of QF's lounge at SYD T1, on my flights to NZ with QF, pre Covid.