Thanks pandaperth. That is what I thought.
1. Speed of Refund
The TA I used first said to me:
"When claiming a residual refund from the airline it does take some time, on average about 6 months, I currently have an ongoing refund with BA from last November (although BA are the slowest)."
That airlines are not speedy about refunds did not surprise me.

2. No Refund for YQ?
However I was told next:
"Its normally not possible to claim back the YQ (fuel surcharge)."
This did surprise me. Why would airlines keep YQ for flights not taken?

3. Does QR not allow refunds on xONEx regardless of what ONEWORLD says?
I replied that I was considering cancelling my LONE3 ex-CAI and applying the refund to a new DONE3 ex-CAI.
This is when I was told:
"the Airline ticket rules on your booking confirmation say:
After departure:
All flights are non-refundable. No refund value if part flown."
The ticketing airline is QR. The flights flown were also on QR.
I questioned the non-refundability some days ago and still have no reply.

4. Can one Cancel an LONE3 and immediately buy a new DONE3 with the refund proceeds?
I was given a quote for upgrading my existing LONE3, but at that time (about two weeks ago) a new DONE3 ex-CAI plus the cost of returning to CAI was not that much more than the refund I calculated I should get from cancelling my existing LONE3.
Upgrading my existing LONE3 based on October 30 prices cost almost as much as a whole new DONE3 ex-CAI in mid-November. In other words, the additional cost for a new DONE3 purchased using a refund from the LONE3 would be far less than the cost of upgrading my LONE3. That is why I am trying to find out the ground rules so once I can pick a date to go to CAI; I can arrange the exchange if the cost still seems favourable.

Is there any rule or procedure that means the calculated expected refund cannot be used toward a new ticket?
If one had to wait for the airline to process a refund, that would be a deterrent.