No, you don't earn the Plus Points twice.
If you earn status via PQP/PQF, you will earn the requisite Plus Points according to the level(s) you earned. If you do not earn Premier 1K status during the year, you will receive 280 Plus Point as a benefit of your lifetime status. Technically you do earn more points by flying, since you get the 40 points for Platinum as well as any additional milestone points (2 per 3k PQP beyond 22k).