No, you do not need 1,050 qpoints. Moving up through the tiers is based upon how many qpoints you accrue in a rolling 12 month period. Each time you reach a tier threshold within that rolling period you will be upgraded and your new tier expiry date reset to 12 months from that date.

So it's perfectly possible to go from burgundy to silver to gold to platinum in a few weeks if you accrue 600 qpoints that quickly. And you can also double dip on the way up, 40 qcredits for hitting gold and then another 60 when you reach platinum. Bear in mind though that some of your travel may need to be QR ticketed and operated, you can't do it all on partner airlines and codeshares. I can’t remember the exact numbers (4 sectors to renew platinum rings a bell?). Perhaps someone else will chime in...

And yes if you buy an AA issued ticket but with your QR FF number on the booking, you will earn in QRPC according to the AA rates for the sector, fare bucket and your QRPC membership level as indicated for AA flights in Qcalculator. Those may or may not be lower than booking a QR ticket, but be aware that many of QR's special offers for earning avios and qpoints require the flight to be both marketed and operated by QR, so codeshares won't earn as much.