Buy two 1-way tickets SYD-AKL to position to AKL; ‘forget’ to take the final SYD-AKL flight on the LONE4 ticket – thus no need for the return from NZ at the end (you’ll be having your NZ holiday at the beginning, right?)

Book the first flight of the LONE4 with LA, which has a daily, very early morning flight AKL-SYD. LA does not charge ANY fuel surcharges, whereas Qantas charges lots

You are visiting LHR twice; I suggest you make the second visit a transit (less than 24hrs) and that way you’ll avoid the high UK departure tax (called APD and is GBP85 for an economy flight to SIN)

You have a number of flights and two short surface segments in the US, using up 7 segments: …HNL-LAX-EWR,LGA-DFW-FLL,MIA-DFW…

This seems wasteful of segments to me. Unless you have a real need to visit DFW twice and to have the surface segments. I suggest …HNL-LAX-JFK-DFW-MIA…, saving three segments which can be used elsewhere.
Similarly with your MAD-BCN segment; drop BCN and use surface transport thus saving another segment
Possibilities for the freed up segments:

• JFK-BGI-MIA and/or other Caribbean destinations
• MAD-AMM-LHR or other European destinations
• Break the LONE4 in SIN, fly el cheapo back to Aus, and sometime later fly el cheapo back up to SIN and have an Asian holiday such as SIN-BOM-BKK-NRT-SYD-AKL

You won’t earn a heck of a lot of miles from a LONE4, so IMHO they should only be a secondary consideration, but…
If you fly the SYD-HNL on the AA codeshare, you could join American Airlines AAdvantage program and sign up for its Platinum Challenge (there is a thread on this in the AA forum)

• It costs ~USD200 to sign up; the challenge is to earn 10,000 elite qualifying points on AA-coded flights in 3 months; a LONE4 accrues 1 EQP for each mile flown, so you would complete the challenge with the SYD-HNL-LAX-NYC flights
• From then on you’ll have Platinum status, giving you 100% bonus miles and lounge access amongst other benefits
• AA miles are generally better value than QF points, for example 37,500 are needed for a 1-way economy from anywhere in SWP to anywhere in Nth Am and NO fuel surcharges; QF wants 48,000 and $180 in fuel surcharges just for a SYD-LAX 1-way!!