Can't help with on the ground NZ questions, but for registering your own domain (which I'd highly recommend - much more professional looking) these folks are the cheapest I've found:
https://registry.domains4less.co.nz
You can host with them (the smallest plan should do what you need) but it might be worth dropping them a line to see if they offer push button installation on wordpress, so that all you need to do is find a template and add your content.
Please let me know if I can help - folks here have been super helpful to me, and its nice to be able to pay it back!