Yes you certainly can. And provided you fly in a fare class that earns 1EQP per mile, then that one journey will complete the challenge. Points to note are:

1. You cannot fly any old airline; it must be one of the airlines on AA’s list of airlines eligible for the challenge (the list includes AA itself, BA and QF)
2. To complete the challenge in this one journey, the fare class must earn 1 EQP per mile (or better); flying AA or BA this is quite easy, but less easy with QF where you must be in one of the higher economy fare classes to earn 1EQP per mile (see Qantas Earning Page)
3. Since you’re planning to fly in September, your status will last until 29 Feb 2016
4. Which means all your flights on the DONE4 will earn 100% status bonus in addition to the 25% class of service bonus
5. And no doubt you’ll want to maintain your status by earning enough EQP on your 2015 DONE4 travels (you’ll need 50K EQP, which means flying 33,334 miles in Business) – with careful planning of your 2015 itinerary you can achieve this (and thus your status will last until 28 Feb 2017)

Yes.
AMM-SYD is Middle East to South Pacific. Miles required are 30K/45K for Economy/Business (see the Award Chart)
You will earn approx: 10K for SYD-LON + 90K for DONE4 = 100K (90K = 40K + 100% + 25%)
Note: not all flights in Nth America are in F, only those in the US on flights that have no business class. So ALL your Nth Am flights will be in business. The difference is not great – a 50% bonus instead of a 25% bonus
Note: the general rule on using miles for upgrades is that you can only do it on your frequent flyer program’s flights. In your case you can only use AA miles to upgrade AA flights (so you cannot upgrade yourself on a QF flight). One difference is that AA miles can be used to upgrade BA flights.

No, as previously explained you cannot use AA miles to upgrade QF flights.

Suggestions:

1. DFW-ANC-DFW. This is seasonal; this year flights commence in late May. When you come to book these flights may not be available (2015 flights not yet loaded into the system) in which case you could leave two spare segments of the 16 allowed and pay the change fee to add them when they become available. These flights will be in 1st class so will get a 50% bonus.
2. LHR-CDG. Firstly you have Paris (CDG) not Rome (FCO); you said Rome in your description. LHR-CDG is a very short and therefore wasteful segment. Perhaps fly ORD-CDG and get the train to/from London?
3. For mileage earning ORD-AMM-LHR is better than ORD-LHR
4. The following 2015 part of your DONE4 will retain PLT status for you:
   SYD-NOU-SYD-WLG-SYD-JFK-YVR-DFW-ANC-DFW-MEX-ORD-CDG-AMM

I suggest more playing with mileage monkey

If the f*$&#$g online worked properly, then you could do the booking online!
AA has a general sales agent in Cairo; search this thread to get details. Yes, phone the AA RTW desk, tell the agent you’re going to pay in Egypt in Egyptian Pounds (otherwise it will be priced at the US price!)

Gardyloo gave you the details while I was typing