IIM

The fee for new IMMs ranges between Rs 4.25 Lacs to Rs 5 Lacs/year. For example, IIM-Raipur is charging Rs 5.13 Lacs while for Shillong, it is Rs 5 Lacs, " They said. IIM-Amritsar, that was announced along with IIM-N, would be charging even lower at Rs 4.5 Lacs, though its admissions are yet to begin.

96+ overall is required for getting calls from baby and new IIMs (sectional must be above 80 percentile). For older IIMs 98+ percentile and a really great GD/PI will do the trick.

Getting calls from IIMs not only depends on academic profile and CAT score but also on co-curricular profile and work profile also. IIMs try to keep diversity in the batch, so they may prefer students from non-engineering background with slightly lower academic profile but a good CAT score. A good score in CAT around 98-99 percentile will definitely enhance your chances to get calls.

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.