...does DOT apply to all legs both to and from the USA or does it only apply on internal USA flights and the return from Houston to NZ ?
The entire ticket, i.e. CHC-MCO, even IAH-MCO will be operated by UA.
And I'm guessing given they were booked via a TA my discussions likely start and end with the TA.
Yes...TAs can be tricky in term of getting refund, as your refund should directly come from the TA, not NZ (but NZ should provide necessary waivers for TA to issue refunds).