M.Sc Biotechnology

https://www.shiksha.com/university/university-of-madras-unom-Chennai-23679/admission
This link will help you.

Taking up a Ph.D will certainly boost your career prospects. There is no need to qualify NET exam for the same. You may need to appear for the entrance exam conducted by the institute. After M.Sc you will find opportunities in research laboratories, pharmaceutical companies, chemical industry etc. If you want to make a career in the research field you must appear for the NET+JRF exam. Normally candidates join any of the research firm as research assistants and prepare for the NET+JRF exam. On clearing this exam they are eligible for the pay scale under the UGC norms. After this they pursue a Ph.D along with their job.

You should write PGCET or go through management seat.

PGT stands for Post graduate Teacher. And eligibility criteria for a PGT is a Masters degree (in any discipline) from a recognised university, B.Ed or equivalent degree from a recognised university. So, as per your question, you have completed your Masters in Biotechnology, but for a teaching post like PGT you need to complete your B.Ed as well.

http://stellamariscollege.edu.in/
Refer this link.

Hi,
You are eligible. You can apply for admission.

Yes, but from what I understand placements are not assured or regular.