Your question is quite complex so I’ve split it up between two topics:
The JAL website doesn’t tell the full story about award availability. Airlines usually release more seats to their own members so if you are using your Qantas points, you likely won’t have access to access to as many seats. If you are keen on using your Qantas points, you should search for seats on the Qantas website. This guide is very helpful in searching for seats in general, along with this guide specifically for Japan Airlines redemption.
Similiar story goes to other programs like Asia Miles. If you want to redeem Asia Miles for Japan Airlines seats, you should search on their own website for the most accurate availability.
Lastly, I’d note that 7 seats even in Economy is a rare find. You may need to split the party of 7 people into 2 separate booking since some only let you book 6 people into a booking at a time.
Frequent Flyer Program Choice
To use your frequent flyer points on Japan Airlines flights (or any other flights for that matter), you should always call or use the website of your frequent flyer program. Japan Airlines can’t help you redeem your Qantas points/Asia Miles, etc.
As to what points to use, you will use the least with Qantas points. You need 31,500 Qantas points for an Economy one-way on Qantas flights or 37,800 points for partners like Japan Airlines. The same flight on Asia Miles is 27,000 Asia miles (transferred from 54,000 AMEX points). Your suggested method with converting to JAL may give you more availability and more seats but you lose out on a lot of points with this method and is definitely the most expensive way. I do not recommend this.