Hi & Welcome to NZ.
You should have gone to the Boat Show at the weekend. All your questions answered in one spot.
There is a Spearos section on this forum.
I just happen to be online and I'm not a Spearo or Diver of any sort but I'm in Titirangi and can confirm there are no shops nearby since Stirling Sports ceased their fishing section in Lynn Mall.
You have Norcross Fishing World in Henderson (Railside Ave.) and Hunting & Fishing out at the Westgate complex as nearest.
Otherwise there are several shops down in Westhaven.
I don't think you can dive for Paua with SCUBA - only snorkels (But I may be wrong).
Google the MPI website. (Ministry for Primary Industries).
You would need someone experienced with you to go near the Manukau Harbour.
Enjoy your visit to 'Godzone'.
Someone will be along shortly with proper answers for you...