bzcat

On connecting flights with same PRN, you should be able to recheck your luggage at TBIT before leaving the secured area so if you can book your NZ and UA flight on the same ticket, this maybe the way to go.

If you are booking separate flights, you may not be able to recheck your luggage at TBIT (it depends on if the relevant airlines have staff at the recheck station when you show up) so you should think about the logistic of walking vs. shuttle bus ride. NZ to UA transfer is a 20 min walk or a 10 min shuttle ride (TBIT stopping at T4, T5, T6, and get off at T7). NZ to DL transfer is a under 10 min walk (TBIT to T2 walk clockwise direction - turn left after exiting TBIT).

If you walk, you can keep the free luggage carts from TBIT custom area. If you take the shuttle, you have to abandon your free cart at TBIT curbside, load the luggage on the bus, unload the luggage from the bus, take it inside the terminal.

Basically, if you can book a single NZ or UA ticket AKL-IAH, that is the best. If you are booking two separate AKL-LAX and LAX-XXX flights, I prefer DL in this case since the transfer at LAX is simpler and walk is shorter and you will have luggage carts, and it gets me directly to AUS rather than IAH.

Edit: booking separate tickets on DL means you have to pay checked luggage fee... so another thing to consider. Staying with UA at least your *G status comes into play on luggage fee.