Murf, LAX-HNL is a domestic route. AirNZ cannot sell tickets on this segment. What they can do is sell them to people who flew AKL-HNL. Here is an example. If someone in Tel Aviv wants to take a trip to Miami and NYC, they can fly El Al on TLV-JFK, then stay in NYC for three days, and then fly El Al on the JFK-MIA segment. Some intra-US routes:
South African Airways FLL-ATL
El Al/NAA MIA-JFK
Aerolineas Argentinas MIA-JFK (coming back soon)
ANA ORD-IAD (ends this week)
Korean Air DFW-ATL (suspended for now)
Qantas JFK-LAX (last flight is this week!)
British Airways CLT-BWI
Not domestic, but Royal Air Maroc flies JFK-YMQ, Cathay Pacific flies YVR-JFK, and El Al flies LAX-YYZ.
In the past, Varig has flown ATL-IAD and MIA-MCO, Lufthansa has flown CLT-DFW, British Airways has flown IAD-MIA (Concorde), LanChile JFK-MIA, and Egypt Air JFK-LAX. The list goes on and on.
Also keep in mind US carriers also operate routes in between other countries with full passenger carrying rights. American Airlines flies Sao Paulo-Asuncion (extension of MIA-GRU) and Buenos Aires-Montevideo (extension of MIA-EZE). United flies Buenos Aires-Montevideo (extention of ORD-EZE) as well. Both United and Northwest have a large presence at Tokyo Narita flying many routes out of that airport. United also flew, just prior to 9/11, HKG-SIN. That route is suspened through Spring, for now.
As for Heathrow, from US carriers, only UA/AA and from UK only VS/BA, as everyone knows. This does not exclude carriers from other countries. Air India flies JFK-LHR, for example. Not every carrier can, though. SQ wants JFK-LHR, but has had no luck.
Carriers with 5th Freedom in the United States include Air France (from MIA), Iberia (MIA), Air India (JFK), AirNZ (LAX), Qantas (LAX), Martinair Holland (MIA), Surinam Airways (MIA), LanChile (MIA, LAX, JFK), Varig (LAX), Cathay Pacific (JFK), Singapore (ORD, LAX, SFO, EWR, JFK), Kuwait Airlines (ORD, JFK), Royal Jordanian (JFK), Ethiopian (JFK, IAD), South African (JFK), and a few others.