CBSE 12th

There so many colleges and universities for engineering. Which gives admission without JEE exam. First of all choose your destination course and search about it on the internet. Then decide and search the best institutions or colleges for the same. Visit the link below for engineering colleges
https://www.shiksha.com/b-tech/ranking/top-engineering-colleges-in-india/44-2-0-0-0.

Hello Aditya,
Please visit the link below for the best B.Tech colleges in Mumbai. As you are willing to do software engineering so this link provides you the various colleges list in Mumbai for CSE:
https://www.shiksha.com/b-tech/ranking/top-computer-science-engineering-colleges-in-mumbai-all/46-2-0-10224-0

Sorry, but all NIT admissions close down to at maximum All India rank of 60000 for freshers batch. Rest rankings are for assuring students, their seats in JEE Advanced. Also, for any NIT, the minimum rank required for CS/IT is 15000 all India. Kindly go for state government autonomous institutes with home state quotas or private universities.
Best of luck.

Kindly refer to my answer on the same question:
https://ask.shiksha.com/with-186-marks-in-jee-paper-2-and-nata-score-97-can-i-take-admission-in-mnit-bhopal-i-belong-to-general-category-qna-4807625
Note: If you are not eligible with home state quota, then it will not be possible at least till the 4th round of counselling.
Best of luck.

You can get a seat but you have to appear in counselling procedures which is 70% seats are filled through university-level counselling based on marks obtained in class 10+2 and rank obtained in JEE Mains. Selection parameter and weightage:
10+2 marks - 75%
JEE Mains - 25%
Remaining 30% seats are filled by the university based on the marks obtained in class 10+2.

For CBSE, at least 75 %. For other boards, refer to the last year cutoff.

Yes, you can get a seat here as the cutoff remains near 120-130.

According to the IIM admission criteria, candidates appearing for CAT 2017 must have a Bachelor's Degree with at least 50 per cent marks (for General category) and 45 Percent (for SC, ST and PWD category) or equivalent CGPA from any recognised university. The minimum CAT percentile one needs to get a call from IIMs is 93 percentile. The percentile includes the marks scores in 10th, 12th, graduation, work experience, gender, etc.