As someone who was exactly in your shoes back in 2020 I think I can answer this question. I was flying a United economy award from Toronto to London Heathrow (via Dulles) as a Premier Gold with myself and my parents. How this works is you'll be able to choose economy plus seating for yourself and one other passenger. For the 3rd, 4th and remaining passengers they will have access to non economy plus seating (including preferred seating). So whilst they may not get those epic over-wing seats that offer 55+ inches of pitch, they won't be stuck in the middle seat either. That being said, I noticed that passenger 3 and 4 are kiddos and per a recent ruling by the
US Department of Transportation United is obligated to seat them with you at no additional charge. How this will work in reality is after you select your seat for yourself and your spouse online, you'll try and have your kids seated beside you. If it tries to charge you for the privilege, call up United and explain to them that they are kids and must be seated together with you for free.
In terms of lounge access, which I suspect is gonna be another question you'll want to know the answer to: you should be able to guest your wife in and the kiddos should get in for free. In terms of which lounge to visit at LAX, it's really a judgement call. I personally prefer the Star Alliance lounge at the Tom Bradley International Terminal (TBIT). They've got showers, great food selection and a really wonderful terrace with fireplace and bar. The only downside here is that United flights from LAX take off from terminal 7/8 which is about a 30 minute walk air-side from the TBIT. The United club at Terminal 7 is of course another possibility. The food ain't as nice and they don't have showers but they do also offer a terrace.
In terms of Sydney airport, well you've got a couple options here. Singapore and Air New Zealand lounges should both be available to you, your partner and kiddos. From my visit back in July both are quite good lounges but I would frankly recommend the New Zealand lounge if it's open. It tended to be less crowded when I was there and had quite a nice food spread indeed.
-RooFlyer88