I will start my reply stating a fact: 777 fleet have no roster "yet".
QA just received 3 planes, large number of training flights, so minimum flights for released crew.
The roster will take a shape eventually; your guess is as good as mine at this stage.
The second point: you want to commute from DXB to DOH?
You must be working for a "normal" airline!
Try to get onboard with ID 90 on QA!! it is a disaster, even with "previously confirmed" annual leave tickets...people were offloaded while the clear print on the ticket said: DO NOT OFFLOAD.
What kind of roster do you expect? you are actually hoping for many days off??? you might get them while in a layover!
When the 777 will work in full force, I expect you to have no time to travel anywhere except in your annual leave.
Ask the 330 crew how it was when they were short of crew? I think it will be the same.