Welcome to FT adamkps

As you are travelling on DJ BNE-SYD you probably wont have lounge access, unless you pay to visit DJ's Blue Room (or have other lounge access through Diners or ...).

At SYD I would use NZ lounge. It is superb - great food and drink, showers, computers, decent size, etc.

My understanding is you will have access to RCC at LAX - provided either your *G status is with airline other than UA and US, or the LAX-PHL is same day connection from the international flight. Coming back is international flight so definitely access as *G. Similarly for PHL on way back (note I dont know what lounges are available in PHL).

On the return through SYD you wont have access to *G lounge as they are only for departing pax. However you can pay (A$5 IIRC) to use DJ's Blue Room which reportedly has showers.

Hope this helps.