Welcome to AFF, Azusa.
Captain Halliday's suggestion to use the route map is a great one. Remember that you don't have to fly direct between two cities. In fact, because you're travelling in Business, you'll ideally want to connect as many times as possible, just to get your money's worth. You won't be able to get to PRG out of Asia directly. Consider routing via DOH or travelling into MAD, HEL or LHR and connecting from there.
I think DAD is only served by KA which may make finding flights difficult. You may need to consider flying into SGN or SIN and taking separate flights as those ports are better served and, therefore, you stand a better chance of getting seats.
As mentioned, booked as soon as seats become available. The downside is you will be up for the change fee of 5K per person, per occurrence (not per change, just each time you make a change or changes). And as you are looking at departing in late August 2020 for six weeks, you won't be able to book everything before the cost of the award goes up. Expect to ultimately pay the new, higher price, in addition to the change fees.
This is increasingly becoming a YMMV. Until Qantas better educates its staff, you just have to hope that logic prevails and SFO/LAS is viewed as one single stopover.