qatar flights disappearing from DONE 4
 

I don't think you should be too concern. It could be a system glitch (or update on the source). Check back later you'd probably find everything ok. Hold on to your Record Locator (or confirmation booking code).
Last resort, still call the AA RTW desk.

Interesting. After your OP, I checked our AA-ticketed DONE4 which has two remaining segments, in December, both on QR. Both appeared in the system, as per normal, nothing to see here folks ...

So, who knows what went wrong with your booking? but, thank luck, it's all fixed now.

Amazingly this is happened again. Only this time three of the four Qatar legs have disappeared instead of all four. I’m beginning to believe someone at Qatar Airlines does not like me.

Could it be due to married sector logic?

This happened to me several times with QR flights in a AA ticketed DONEX, including one time 1-day before the QR flight where there were now no business class seats available. Thankfully, the AA team were able to work with QR and get my wife and I on our originally booked flights. I'm not sure how or why but I'd caution anyone/everyone to double check QR flights, especially with AA plated tickets. I don't have enough exerience to know whether the issue exists outside of the AA ticketed / QR flight combo, but it happened 3 of the last 3 DONEs.

Unlikely; "Married Segment" logic influences availability on booking, not once ticketed.

Just a data point:
My currently in progress DONE3 was ticketed by CX. Among the remaining segments, of which 3 are QR flights. My record is still intact. All QR segments are showing in AA, CX, JL, QR, MH web sites.

Just checked the remaining flights on my current itinerary as well as the two I haven't started yet. Nothing missing.

I don't think this is specific to QR, or that QR has anything against certain flyers. My guess is that it's a glitch stemming from an itinerary booked and ticketed in one GDS (such as Sabre, used by AA) not correctly propagating ticket numbers to a separate GDS (such as Amadeus, used by QR, QF, CX, BA, etc.) used by the airline operating the flights. I've had this happen on AA-issued xONEx with QF flights. In my experience, the most likely trigger is when a change is made, causing the tickets to be reissued, and the new ticket numbers don't get correctly transmitted to the operating airline's system.

If you want to be extra-safe, after an itinerary is ticketed or a change is made and the tickets are reissued, call any airlines with flights on the booking and ask them to verify that they see the itinerary as ticketed. If they do, you are fine. If they don't, call the issuing airline and have them resend the ticket numbers.

Yup, I had this happen on an AA award that included QR segments. I made a change in routing about a month out. Everything looked good until I showed up in MIA and they wouldn't let me board. The GA spent the next 20 minutes on the phone with AA getting them to re-issue and re-ticket the remaining segments of the trip. Ended up being the last person on the plane.

oneworld circle PNRs are supposed to have an OSI message in them suppressing the autocancellation

But it probably doesn’t work.

In any event check in would not be possible if the PNR no longer had a reference to a ticket coupon that was open for use