M.Tech

Go through the syllabus or content of these options and the one which will suit you the most in achieving your goal, select that option.

You will qualify the GATE exam but the score will be quite low.

You have to score good marks in GATE exam then only you can get admission in any good college for M.Tech as you have scored less, you can take one more chance, prepare well and take GATE exam again next year so that you can score good marks.

To get into a good IIT. You should get a minimum of 700+ in GATE.

Please go with this college predictor, this will help you to choose the best for you:
https://www.shiksha.com/college-predictor.

To teach in schools like KV you have to also clear the ctet/tet exam. After that you can apply for the job.

The best M.Tech specialisation for mechanical in terms of job prospects at JAIPUR NATIONAL UNIVERSITY. M.Tech ME (INDUSTRIAL & PRODUCTION engineering. )
Why JNU Jaipur? Experienced faculty
State of the art laboratories
Research articles in reputed international and national journals
Junior research fell

Yes, you are eligible but you need to score a good rank in GATE for colleges like IIT & NIT. For BITS, there is a separate entrance exam for M.Tech.