dolfinco there are several lounges at DOH. Look at the Wiki on the QR dashboaed widget
QR Lounges at HIA to see which are available.
Since you and your wife are both travelling in J (Business) you have access to Al Mourjan lounge. But as you will see guests are not permitted..
As a BA Silver (OWS) you arre also able to use the much poorer Business class lounge where guests are permitted.
With regard to SYD you will be able to use the Qantas International Business lounge where guests appear to be admitted. Further details of lounges can be found on
https://www.oneworld.com/ffp/lounge-access.