GATE

You can try in any CSIR.
Good luck.

Yes you can get. Please use Shiksha college predictor.
Yes, if you cleared cutoff with good marks then you definitely will get the stipend. So, you can apply for admission. Good luck.

Here are the complete details of GATE EXAM along with eligblity parameter of various qualifications
https://www.shiksha.com/engineering/gate-exam.

No, you are not eligible for M.Tech from BIT K R Puram without appearing in GATE exam. Admission in this college is done through GATE ranking, which will create problems for you to get admission. Best of luck.

The minimum required score for GATE is 650.

All private universities and colleges generally don't require any GATE scorecard. So you need not worry but carefully select a college that has a good placement record and studies.

One might be pursuing a Master's course in any branch of Science. Mathematics, statistics, C. se or equivalent courses and can attempt the GATE exams. However you should be in your final year or have completed your course in 2018. If you are supposed to complete your course in 2019 later, then you would require a certificate from the principal instead of your pass or provisional pass certificate.

No, qualification required is GATE.
https://www.shiksha.com/engineering/course/m-tech-delhi-technological-university-rohini-230259

What is your branch? This is important and different PSU having different cutoff marks and rank. Like for mechanical the rank up to 1500 sometimes.