Your questions about the company - my remarks there were based upon what people have sometimes said on our threads, how INZ wanted to check the business out to make sure, in effect, that it would still be likely to be there for the next few years to employ the migrant to whom they were offering a job. They asked about the firm's finances and business plan. There is nothing in the written regulations to say what triggers such an investigation. The requirements for a skilled job are listed under the various headings here, and that's what you will need to go by.
http://www.immigration.govt.nz/opsmanual/30545.htm
About submitting proof you have a job - you don't need to work in the place for three months before letting INZ know. You can contact them as soon as you have the offer.
http://www.immigration.govt.nz/opsmanual/30614.htm