Be aware that JFK has about 9 different terminals. If the lounge that permits access based on Amex membership is not in the terminal you are using, you won't have access.
If flying with AA out of Chicago (ORD) a Qantas Club member has access to the AA Admirals Club lounge.
Helsinki (HEL) only have AY lounges which are not accessible by Qantas Club members.
Depending on your routing, it may be worthwhile signing up for the AA Platinum Challenge. That way you will earn OneWorld Sapphire status and have access to OneWorld lounges when flying with OneWorld airlines. That will provide access to AY lounges in HEL, as well as AA/BA lounges in LHR (depending on which airline you use for LHR-ORD), AA lounges in ORD, SFO providing on a same-day international flight with OneWorld airline). It will probably permit access to the QF partner lounge at NRT (JAL lounge) but might need to check on that one since its really a partner lounge). And certainly access to the Qantas Club lounges in Australia.
Assuming you are booked on QF107 SYD-JFK (transit in LAX, but as a single flight number all the way to JFK), you will have travelled 9950 miles. Depending on the booking class (and assuming economy travel) you will earn either 4975, 9950 or 1,495 Elite Qualifying Points towards the AA Platinum Challenge. You need 10,000 points to make it to Platinum.
So you need to find out the booking code for the flights (H, K, W, M, V, L, and R all earn 1 point per mile; G, S and O only earn at 0.5 points per mile; B and Y earn at 1.5 points per mile).
So if booked in H, K, W, M, V, L or R, its probably worth adding a domestic flight in Australia first (say SYD-CBR, SYD-NTL or similar) to earn 500 or 1000 points before you leave. Then you also earn the 100% mileage bonus for the SYD-JFK flight, which is a great bonus. And from then on you have lounge access based on AA Platinum/OneWorld Sapphire status.