MrGatch welcome to the QR forum. You will find that the dashboard is a useful tool when accessing the forum.
1. All wide bodied aircraft, and a few narrow bodied have lie flat seats. The use of cabin class F has nothing to do with the seat type. There is no J on most regional flights, but F and Y only. You could have an A320 or A321 with cradle seats and it would still be an F flight.. I personally can not see the need for lie flat seats on this short hop DOH-DXB. No sooner than you can lower the seat after take off than it must be raised again for landing!
2. Prior to the DOH-DXB flight you will have access to Al Safwa because you hold an F boarding pass. On your return DXB-DOH you are not arriving on a longhaul flight and do not qualify for access til Al Safwa but would to the QR premium lounge in DXB.
3. With the exception of the A380s QR has no three class aircraft! I very much doubt QR would use its overstretched A380s for a regional flight. They are all needed elsewhere. Thus there is no chance of purchasing an upgrade.