Hi and welcome to the forum,
1. Unsure as this isn't my field of expertise, but it is a skill shortage on the LTSSL, therefore there should be jobs out there.
Try and to search for jobs which will also show the names of the recruitment agencies dealing in those types of roles.
2. Yes you can search jobs from India and apply from India. If you are a person of interest you could be interviewed over telephone and/or Skype. You can't make an employer in NZ give you an offer letter. All you can do is hope to get a formal offer in writing.
3. The visa is Residency Visa not PR
. Common misconception. You can only get PR
after you have been a Resident visa holder 2 years. In answer to your question - No, you can only go to Australia when you have gained NZ citizenship which can only be applied for after 5 years as a Resident visa holder.
4. Your parents can migrate to NZ if you get Residency but only after you have lived in NZ for a minimum 3 years as a Resident and you satisfy the criteria to be a sponsor - you have to earn enough to support them both etc etc.
5. As you have said, you need a job offer to enable you to claim enough points to apply for Residency via Skilled Migrant Route. You also need a job offer in NZ to secure a Temporary Work visa or Work To Residency visa. The only visas that will allow you to come to NZ and search for work are the Working Holiday Visa and the Silver Fern Visa. Look at both via the Immigration website as you have to meet the eligibility criteria for both in order to apply.
6. If you can't prove that your study was taken in the medium of English then yes you must have IELTS pass at minimum 6.5 overall in order to meet the English Language requirement for all visas into NZ.