I'd suggest using OS to get to DXB via VIE - they will be the carrier that can take you DXB to Male nonstop after your visit to DXB, so there's a certain logic in keeping the three segments on the one carrier.
(I say this as I'm not sure of the status of DXB-MLE flights on OS - they are permitted on the RTW product, but I'm not sure if they can be "sold" just from DXB without an arriving International flight on OS). The other alternate is LH through FRA - I think the MUC flight to DXB (which has better timings than the FRA flights) ceases over the Northern summer.
From Male you can get to SIN with SQ which is also a good choice as well as a logical one.
Take your pick between NZ and SQ between SIN and N.Z. - NZ has good comfort/service levels in Y and SQ has the personal entertainment units at the seat as their major respective "plusses".
Not much choice to get from N.Z. into Nth America except using NZ, but as mentioned before, their Y product gets good reviews so it's not a loss in using them.
Take your pick trotting around Nth America - US UA AC - they're all much the same domestically. Go with the carrier that either wastes the less miles for you (if you're strapped for miles) or the one that can add the miles for you (if you have the miles to spare and want to earn more for the FF account!).
Beware of Ted's, Skidoos and Jungle Jets however.

Cheers.