Going Boeing -
The information I found is bags are tagged the whole way through for QF codeshare customers through SIN. FF Points are earned in the top five fare brackets for codeshare customers YBHKW (Qantas marketed operated by Jetstar)...
The seat pitch is a bit rough when you consider most codeshare pax would be connecting to or from another flight as part of a long journey. It isn't just the CNS/DRW-SIN on its own.
What Cairns does need is more full service carriers like Singapore Airlines, Malaysia and Thai Airways
All the above are yet to return to Cairns after some (in some case many) years. They all have massive markets to offer customers ex their hubs yet still deem CNS unviable. Qantas has a much higher cost base and fewer connection opportunities to offer ex-SIN making its ability to earn money even more limited. What Qantas need is some appropriately configured 73Hs (or 739ERs) for such markets (CNS/DRW-SIN, PER-BKK etc), in my opnion.