1) IELTS is not mandatory if you can produce a letter from the degree awarding university that your entire course of study was in English. However, nothing stops a case officer from asking for a IELTS report(a CO is assigned some weeks/months after you submit your ITA application, and ITA is after your EOI getting selected). If, in the interview with CO, you communicate well, then the CO may not ask for a IELTS report to be seen, but watch the word 'may be'. No need for IELTS at EOI submission stage. However, IELTS wait times are long, do check and schedule something just in case. But it costs money, and as I said, it entirely depends on the CO. On this forum, to my knowledge, and do note, my knowledge is limited, I've not come across a situation where CO demanded (insisted) for IELTS report despite the candidate producing a University letter. There are opsmanual guidelines on what basis English requirements can be satisfied. Do look for that on INZ immigration website. I assume your points calculation to be correct.
2) I do think this will have to be issued in India. Not sure if the doc you have will suffice.
3) From what I understand transcripts and individual marksheets are the same. However, you may need a University certified syllabus, but this again depends on NZQA (if you need your quals to be assessed by them)
4) This is not true, as far as I know. The residence visa is valid for a total of 2 years from the date of activation unless you get into crime and other malpractices that may invalidate the visa (activation is done the day you land in NZ first time). The 3-month hearsay you mention here may be this: If a candidate claims points from a NZ job offer, then that job must be held for a min. of 3 months in NZ. Not sure where that someone hit upon this 3 month for the situation you talk about. Rather than banking on rumour, best to give INZ website a meticulous and thorough read.
All the best