Can my first cousin who is a New Zealand citizen sponsor my living expenses, and will that sponsorship work for visa purposes?

No, your first cousin - even if a New Zealand citizen - is generally not considered an eligible sponsor for your student visa from India. Immigration New Zealand typically accepts only immediate family members (parents, siblings, grandparents, spouse, or parents-in-law) as sponsors.

You'll need to show you can cover your living expenses independently or have an eligible immediate family sponsor in New Zealand.