1. I never experienced it but I would bet they expire immediately after your Platinum is over and don't stick up to the end of the quarter.
2. Aside of certain promotion, to reach Platinum you need to earn 600 Qpoints in a 12 months period of time (and get at least 20% of these Qpoints from QR flights or from at least 4 QR flights). Even if you don't earn any Qpoints during your Platinum status year (30-NOV-2019 to 29-NOV-2020) you should be able to get a Gold status after that, because you meet the 24th month requalification Qpoints thresthold for Gold, which is 540 Qpoints over the past 24 months (you got 600 Qpoints), provided 20% of the 540 required Qpoints came from QR flights or provided you credited at least 8 QR flights over the 24 previous months (the requirement is 4 flights for a 12 months (re)qualification period and 8 for a 24 months (re)qualification period).