IIM

Yes, definitely, you have a chance to get a call from IIMs, provided you do well in CAT. For a better idea, you can visit the website of IIM and can check out about the weightage each IIM gives to these components, as each IIM gives different weightage to your past academic record.

You need to qualify CAT in order to get admission in IIM. Did you take that exam?

Top MBA colleges in India are mentioned at the link provided below. This is a comprehensive list which contains the Iims as well as other top B-Schools in India as well. This ranking is based on rankings published by The Week, Business Today and National Institutional Ranking Framework (MHRD).
https://www.shiksha.com/mba/ranking/top-mba-colleges-in-india/2-2-0-0-0

Both are good B-School. IMT, Ghaziabad is better in terms of location, infrastructure, alumni network and others. IIM, Nagpur has just IIM tag nothing others.

JBIMS, IMT Ghaziabad, XLRI, VIT, Pumba, Sydenham, Welingkar, Balaji, Symbiosis etc. Are good enough. Everyone have different entrances and the application process starts in August. Start preparing and best regards.

Yes, IIMs provide engineering management.

IIMS Pune is the second best in Pune for MBA after Pumba. Considering ROI, it's better than Symbiosis and Balaji. Placement is excellent and overall growth is there. Curriculum is well designed with lots of extracurricular activities.
Best of luck.

IIMs are the top B-Schools in our country. The Indian Institutes of Management are a group of 20 public, autonomous institutes of management education and research in India. They primarily offer postgraduate, doctoral and executive education programmes. Normal colleges don't provide better faculty members, facilities and placements.

You must write MBA entrance exams. Try to score higher scores. There are many good B-Schools with low fees. You must target for colleges like
Faculty of Management Studies, Delhi
JBIMS-Mumbai
NITIE-Mumbai
TATA Institute of Social Sciences -Mumbai
IIT-Delhi/Roorkee/Kharagpur/BHU
K. J. Somaiya Institute -Mumbai
Pune University.