Kia Ora Bro'
Sorry I didn't get back to you sooner but things are only getting back to normal after a hectic New Year's.
Anyway, I will do my best to answer some of your questions.
Actually, it depends a bit on where in Dorkland you are staying and how much access you have to transport. Jaffaland, like most of Aotearoa is car country. Public transport is OK, but not really geared towards tourists.
It also depends on what your definition of 'cool' is. Getting pissed? gambling? Adrenalin sports? LOTR? Museums? Art? Rugby? You can do a bit of everything in and around Dorkland. There's even two GWs, although with NZ prices and your general disdain for 'em you might wanna give them a miss.
As for ribs, are you ****ing kidding? New Zealand =/= America, ya know? You can probably find 'em but why would you want to? Dorkland is as cosmopolitan as Sydney, you can get pretty much anything from anywhere on the eating front. If you are looking for some kiwi 'soul food' you can't go past a good meat pie. Much better than the shit Aussie ones you can get here. Start with mince and then explore from there.
Anyways, Kojibear can probably be of more help as he is an actual 'Jaffa' while I am a 'Mooloo' man.
As for the things you mentioned in you PM, I've been thinking long and hard about it

The thing with NZ's most iconic songs is that they are often partly or totally in Maori, for example....
Pokarekare Ana.
http://youtu.be/9GXua6gD4Hc
Or
Now is the hour.
http://youtu.be/8Su-O-mfNVI
Actually, the latter would be particularly appropriate if you were doing it at a farewell shindig. The Maori part wouldn't be that hard either, as the language is similar to Japanese, pronunciation-wise and would be easy to write in katakana.
If you didn't want to try some Maori you could try, "How great thou art" which is a popular hymn/standard in NZ or perhaps "the world in union"
http://youtu.be/BGBYpzwlAo4 the theme to the Rugby World cup which is pretty popular.
Well, that's about all I can be arsed writing for now. Give us some specifics about what 'cool' things you and I'll give you a few more ideas.