No idea about using Avios on Qatar, but I will point out that you're in the disadvantageous portion of the multi-carrier reward chart at ~11k miles. If you want better "value" you could book another 8500 miles of flying for your daughter for an extra 10k miles Avios - enough for a round trip to the US, for example. You're allowed stopovers, so this could be at any time later on. Furthermore, the multi-partner rewards price on the cabin of the longest leg, so you could put her in business on all but the Australia-DOH leg for the same number of miles Avios if you can find the space.

The problem with the multi-carrier redemptions is that basically no rules are published by BA. Some of the blog articles about it will say that you're limited to a "return standard", which may mean that the above Aus-UK-US-UK gets nixed. However, there may be an element of luck to that so can't hurt to try. In any event, a "return" leg to Asia ought to be ok. If you have further questions you can PM me.