Hi Luane,
Welcome to the forum
To try and answer your questions:
1. If you are on your own, and want flexibility then backpackers hostels are cheap as temporary accommodation. There are also many places to rent, depending on your needs. Prices will vary greatly depending on which city you want to live in, and in what area you choose.
2. This depends on what type of employment you are looking for. Long term skilled employment is usually on Trademe.co.nz or Seek.co.nz, and with employment agencies etc. Another good option is to contact employers direct.
3. Cell phones aren't a great option for international calls, but if unavoidable it will be cheaper to have an NZ mobile than a SA one.
4. Visa costs vary depending on where you apply. There are branches all over the world, and in most big cities in NZ.
5. Documents required will also vary depending on your job. For work permits, usually some evidence of qualification/experience is needed. For a short term work permit you wouldnt need a medical, but you may still require a police certificate.
6. Again assuming you are on your own, day to day living wont cost you much. The biggest difference to living costs will be your accommodation, but as for food etc, not much at all.
I hope that goes some way to answering your Q's!