Answers
New Zealand citizens have the right to enter and stay in the Schengen area without an entry visa for 3 months (continuously or intermittently within a calendar semester).
If an extension of stay (in Greece) is requested - with an application and supporting documents as required by law - the application is evaluated by the competent authorities regarding the validity of the reason invoked by the applicant.
It is not guaranteed that the application will be approved. Regarding the D visa, it is granted ONLY by the Consulate of Greece in the country where the applicant permanently resides, either a citizen of that country or under a legal residence permit and ONLY for those reasons as strictly defined by the law (working as an executive in a subsidiary of a foreign company, or an employee of a shipping company or as the spouse of a Greek citizen or for family reunification, etc.)