Having flown to AU several times -- here's what I can tell you. You want to arrive as early as possible into AU, before 7AM if possible. 7AM hits and it is a madhouse getting through immigration. Although the AU immigration works are bright and cheerful, it does not necessarily make up for the crush of people coming in right about 7 AM. For example, one time I arrived at 6:20 or so and was in a cab < 20 minutes later (from a Y seat) -- one time I arrived at 7:05 AM and was stuck until about 8:45 AM.

As far as which carrier, you have many choices, especially if you want to connect through NZ (which may be cheaper). I'll assume you don't want to do that, in which case...

1- Do not fly United. The service has declined severely in recent years/months, and although the 777 and 787 are an improvement (for Y anyway), they are outdone by the other carriers. Business class on UA is OK, but is still outdone by everyone else.

2- If you want to stick with a US flag carrier, DL is well... the only other option. They have a direct NYC-LAX-SYD flight that is reasonably well timed. Y+ will offer you more legroom than regular Y, but if you're not too tall it doesn't matter all that much. The Y meals are passable on DL.

3- The problem with flying DL (and UA) is that the limited number of flights means you're sunk if there's a MX issue. By the time the plane is pulled for MX, it'll be too late and you'll be stuck on either side for an additional day.

4- QF is always good, but is usually much more expensive than DL/UA. Now, you do get more amenities in their premium economy (bar service, better meal...).

5- DL will arrive to T5 immigration on return to LAX. It's a long walk to the CBP station, but it's 100x smoother than going through TBIT. UA will usually use T6/T7 (though lately they have used TBIT occasionally), which is a toss up. If you get through fast T6/T7 is fine, but otherwise it is arguably the worst entry point in the country.

Honestly, if I were you, I'd plan a long connection in LAX for you get a good meal ahead of the flight, and just bring snacks on board. Y+ on DL will give you decent legroom and a reasonably comfortable seat for the lowest price. Make sure you throw away absolutely everything that is edible before arriving in AU, you don't want to have to declare any food.

You should also be able to get access to the DL lounge with your Amex platinum, they have decent snacks in there and showers (which are nice for the early arrival back in LAX).

That's what I'd do.