Blow off the bike cover. Get good rain gear, you will use it. Jacket, gloves and shoe-covers are key, pants are not really that important - a pair of warm tights is fine, or you can get tights that are sort-of wind/waterproof - that's what I use. Something like http://www.rei.com/product/744302
You should get a jacket you are comfortable riding in, and that has pit-zips, is fairly long, and doesn't billow too much, while still being able to put several warm layers under it. I used this http://www.rei.com/product/746520
for a while, but decided that it is really really important to be visible - so go ahead and get a screaming yellow jacket with reflectors all over it. You're going to be weaing it in low visibility conditions, and it will be the biggest thing that drivers might be able to see.
I found it useful to have a mirror, so I could get off the road when needed - roads there are narrow and don't have a lot of traffic, but there are big trucks sometimes and the roads are curvy, so sometimes you have to get off the road to make room.
I found gear (bike parts, clothes) to be expensive in NZ vs. the USA (3 years ago), so make sure you really want to buy stuff over there. (Don't know where you are now.)
In terms of your bike purchase, they build the roads STEEP there, definitely try to get mountain bike gearing (22 granny, 32 or 34 cassette).
NZ is the absolute PERFECT place to solo bike tour for the first time, towns are close together, lots of camping, lots of hostels, tons of friendly people, nice roads, great scenery.
The campgrounds "Holdiay Parks" there generally have a kitchen with burners and fridges, so if you are planning on camping/cooking, you might not need to bring a stove. I brought one and only used it 2ce in 7 weeks - used the camp kitchens the rest of the time.
Your plan sounds pretty good - but NZ is so easy, you don't really need much of a plan. The book "Pedalers Paradise" (there are 2, one for each island) is a very useful resource to get ideas of service locations and hills on almost every road.
Arthur/Porter Pass is awesome.
Wanaka is nicer than Queenstown.
Dunedin is nicer than Christchurch.
From Q-town, you can take the boat across the lake, ride 2 days of remote gravel roads thru sheep station, end up in Te Anau, you can ride up to Milford or bus from Te Anau. Haast Pass from Milford to the West COast is quite nice.
West Coast = Wet Coast
CCH and the surrounding area is the flattest part of NZ.
It's really windy, the wind changes direction 90 degrees each day, you just have to deal. flexible plans help - you might find yourself choosing your next destination based on wind direction.
There's a cool bike-only hostel in a sheep ranch on the Kaikoura Coast.
Eat lots of ice cream, it's really good there.