Mate, we have a couple of poisonous spiders here but they are so rare that if you get bitten by them you will be come a national celebrity! There are no snakes, reptiles or predator type animals.
As mentioned the dreaded sandlfy will cause you irritation but can easily be defeated with any deet based product. Dont worry about bringing in heaps as you will be able to buy it in any tramping or hunting shop. As for tents, bivvys or flys. I'd suggest a bivvy bag, a self supporting fly and a good thermarest type sleeping mat. You might well be camping above the bushline in tussock where there are no trees to secure a hoochie to which is why a self supporting fly or a light weight tent is needed. I have been holed up in the rain for 3 days unable to cross a river, being under a hoochie would have been no fun, it was miserable enough in the tent. The lesson learnt from that, take lots of tea bags as I ran out and started recycling! I have a tent that has a waterproof outer and an inner with built in groundsheet. I leave the inner at home to save weight and pitch the fly which pegs down. I usually carry my bivvy bag and if I am confident that it is not going to rain, just use that under the stars. The milky way in the southern hemisphere with its clean air and lack of light pollution is a sight to behold and can keep you awake for hours just gazing at it.
If you are going into the wilderness you must must must buy a NZ compass, dont bring one from the UK as they dont work properly and you will need to get a 1:50,000 map for the area where you are going, again available from any outdoor / hunting shop.
Check out
www.doc.govt.nz for info on back country walks and hut and
www.mountainsafety.org.nz for info about safety in the mountain funnily enough
There is a fantastic network of DOC huts through out the NZ back country that do have a small cost associated with staying in them, you can buy a season pass for a pittance and away you go. The huts usually have communal bunks (dont get excited, there is no such thing as a sexy tramper!) a bench, wood fire if in the bush and gas if above the bush. 90% of them are fantastic and the remainder have character get in and use them.
There is no drama about carrying a knife/axe openly in the bush, dont do it in the cities though. Dont be surprised if you meet up with people with firearms at a hut, they are hunters and are the salt of the earth. Say hello and introduce yourself.
Now to the deadliest element of NZ. It is the changeable weather the associated risk of hypothermia. I suspect that you will be as prepared as the best of them so it shouldnt be a problem but if you are staying in a hut and decide to go for a day hike (called a tramp or the verb tramping) do not be tempted to dress lightly in shorts and a tee shirt. That is a recipe for getting caught out and at best becoming wet and cold, at worst becoming dead and costing the taxpayer many $1000s in you search and rescue or body recovery! That and river crossings, most tracks have bridges or three wires but on occasion you might be off track and have to wade across. Be very careful, some rivers are deceptively deep, fast and cold enough to make you gasp. Cut a length of sapling and use it as a 3rd leg.
Ohh and finally, make sure any equipment you bring into NZ is spotlessly clean and disinfected especially anything that comes in contact with the ground like boots and tents. Id also leave any wooden utensils you have made at home, they could get confiscated at the border or you get charged to have them irradiated.
Hope you have a great trip, spend lots of money and help our economy
Mac