New story du jour: I printed out my confirmation from QR's website and it also shows Fare Code I (same as AA), which is eligible for advance seat selection. I tried calling QR in Doha. Got a nice agent who told me that the entire DFW-DOH flight has been blocked for advance seat bookings and it will only open up for seat allocation 24-72 hours before the flight! I asked her if I had booked a pure QR flight, would that still be the case and she said yes! She agreed with me that my QR fare code was normally eligible for advance seat selection.
At this point, I think its time to concede defeat. Every time I call, its a different story. I guess, bottom line, all the QSuites are good and even if I get a middle one, its OK.