I'll take a stab at it.

MCO-LAX-LHR on AA
LHR-DXB on BA, stopover
DXB-LHR on BA

If you are inclined to earn extra miles here, you can take another RT from LHR to any country within Europe not on the prohibited list, but I can't think of any flights that have A class.

LHR-HKG-SYD on CX, stopover in SYD (not HKG)
SYD-AKL on LA, stopover
AKL-SYD on LA

Now you have to decide what's more important: visiting both DRW/CNS or CX 1st class.

Option 1: SYD-DRW on QF (biz only)
DRW-CNS on QF (biz only)
CNS-HKG on CX (biz only)
Option 2: SYD-[DRW or CNS]-SYD round trip (biz only)
SYD-HKG on CX (1st class)

Separate RT ticket to PVG.
If there's time and inclination, a RT on CX to some other place in Asia.
HKG-SIN on CX, stopover (some flights have a 1st class section)
SIN-HKG on CX stopover

HKG-JFK on CX
LGA-TPA/MCO on AA.

So far that's 15 segments. Use the last 5 for obscure routing to ANC, like MCO-BOS-DFW-ANC-DFW-MCO.
Or start the trip in MIA, buying your own way from MCO (MIA-LHR). Then you have 6 segments left, and you can use 4 of them on MCO-DFW-ANC-DFW-MCO and use the other 2 for one of the optional round trips in Europe or Asia.