Maharashtra

You should go for Singhad or DY Patil college.

There are some good colleges, accepting MAT scores in/around Mumbai. Some of them are:
ITM Business School, Kharghar.
Institute of Business Studies and Research, CBD Belapur.
Thakur Institute of Management Studies and Research, Kandivali.
Thadomal Shahani Centre for Management, Bandra.
Durgadevi Saraf Institute of Management Studies, Malad.
Vivekanand Education Society's Institute of Management Studies and Research, Chembur.

Well, its good for placements. Although, there is a very tough competition, since all the colleges are together for placements. If you have it in you, you'll surely get placed in a good company. Not only placements, teachers for some of the departments are very good.

This is the link where you can see all the colleges with rating and fee structure:
https://www.shiksha.com/mba/ranking/top-mba-colleges-mumbai-all-accepting-mat/2-0-0-10224-306.
Some college are listed below.

Yes, you can if you have a valid degree certificate, and your entrance exam scores match their cut-offs. Do once again confirm with admissions committee of the college, regarding your graduation criteria.

No, it will not but your score in engineering should be good.

Can't comment on the cutoff this year, last year cutoff was around 93 percentile. And college will rank students as per marks and category ( HU OHU OMS ).

The eligibility criteria for a sub inspector is a graduation degree. Therefore, you are advised to do graduation degree in subject of your choice.