You seem to be late in planning and booking this. Close in flights can be expensive.
For 21 days (3 weeks) you have far too many destinations. You will spend a lot of time just just getting to/from airports, at airports and flying. If you are determined to ago to Asia drop NZ.
1. BNE is long way from the reef. Only about half way from Sydney. You need to look at a map.
2. Do you mean
Great Barrier island? And why?
3a. Many airlines from AKL. Look at the airport web site
3b. Many low cost airlines in Asia
4. A one way fare may/will be expensive. What research have you done?
No advantage in
combining, what ever that means to you
Airlines that fly Australia (CNS* MCY* BNE OOL SYD CBR MEL ADL PER) to/from New Zealand (AKL WLG CHC DUD ZQN) (not from all airports to all airports).
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Air Asia D7 (OOL - AKL)
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Air New Zealand NZ (codeshares with VA) [Star Alliance] {* winter seasonal}
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China Airlines CI (BNE SYD MEL** – AKL CHC**) [Skyteam Alliance] {** November to March}
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Emirates EK (SYD CHC codeshares with QF). EK stopped A380 SYD AKL flights on 13 Jul 2017
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Jetstar JQ (owned by QF)
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LAN LA (SYD AKL) [OneWorld Alliance]
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Philippines PR (CNS - AKL)
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Qantas QF (codeshares with EK) [OneWorld Alliance]
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Singapore SQ (CBR WLG) [Star Alliance]
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Virgin Australia VA (codeshares with Air NZ & AC*) (Delta & AC* partner) [*= some routes in 2017]