Flying QF in business or first gives you lounge access at LAX, MEL, PER and SYD.

LAX-MEL departs from TBIT. There is an interim lounge while the terminal is being remodelled. No showers. If arriving on AA with a moderate length connection, stay in T4 and visit the Admirals Club there (with showers) and head over to T4 at least an hour before departure (90 minutes would be safer).

At MEL you can use the domestic lounge. This has showers.
Ditto for PER.
At SYD you can use the domestic lounge before making your way to the international terminal, and the international business lounge (unless travelling in first or have One World Emerald status in which case can use first lounge) once you have cleared immigration there.

Back at LAX you can use Admirals Club since travelling on an international itinerary in business or first class.