What it checks, and why each one matters
The unit. Brand, model and nominal heating capacity at 7 degrees for every unit, plus the 2 degree figure if your winters get cold. Without those you can't check the size or compare two quotes.
Where it all goes. The spot for each indoor and outdoor unit, and how close the outdoor unit sits to the neighbours. Noise limits live in each council's district plan, and a unit that needs no consent still has to meet them.
Mounting and drainage. How the outdoor unit is fixed, and where the condensate drain (the hose taking water from the indoor unit) ends up.
Electrical work. Who does it, and that you'll get a Certificate of Compliance. Connecting a heat pump is prescribed electrical work, so only a registered and licensed electrical worker can do it.
Refrigerant. Who handles it, and what they hold. There's no government refrigerant licence here. Under an industry arrangement the Commerce Commission authorised in 2011, wholesalers may sell refrigerant only to buyers with an RLTB Refrigerant Licence or an EPA Approved Handler or Approved Filler test certificate.
Body corporate. For a unit title, whether the body corporate has to agree to an outdoor unit on common property, and who'll ask it.
Warranty. The manufacturer's warranty for that model, and what the installer covers on their own workmanship. Your Consumer Guarantees Act rights apply either way, and they can't be signed away.
What a full set of ticks won't tell you
Whether it's really a quote. Consumer Protection says a quote is an offer at a set price that can't rise once accepted unless you agree to extra work or the scope changes, while an estimate is only a best guess. If the paperwork says estimate, ask for a quote.
Whether the size is right. Check it against our size calculator and ask how the installer worked it out.
A seller who knocked on your door uninvited is a different situation: you can cancel an uninvited direct sale within 5 working days of receiving your copy of the agreement. That right doesn't cover an installer you invited round to quote.