IIM

No, there are no direct exams for MBA in IIM apart from CAT. One has to qualify CAT to get admission into IIMs. Apart from MBA, there are some management courses which are launched by IIMs individually.

I would suggest you to go for TAPMI. TAPMI is among top 15 colleges in India and better than latest IIM's. If you are not joining this year, then you must certainly get into IIM-A or IIM-B. It's risky to lose one year and the gain is not that much. I would suggest you to go forward with TAPMI admission process.
Thank you.

Appear for CAT. Score 98+ percentile. Do well in your what PI. Take your seat.

Yes, you have chances to get IIM-Nagpur.

Chances are less to get a call at this score. Slight chances of IIM-Jammu started last year.

BBA is general management studies and MBA is an enhancement of the knowledge. To pursue the career as a manager, BBA+MBA is the best option.
B.Tech is a technical field and students interested in the technical knowledge opt for B.Tech. Students after completing B.Tech work as the technical assistant and to move higher in the organisation they need managerial skills and hence opt for MBA. For B.Tech graduates, MBA opens more options as they have technical knowledge along with some managerial skills, hence benefiting them to understand the business in the better way.

They only accept CAT scores.

If you are from general category, then it will be very difficult as the cutoff will be around 92 percentile for most of the IITs.

CAT is highly competitive. Anybody with a plan and hard work can clear CAT to do MBA in IIMs. First, you must do your strength and weakness analysis. Based on this you can plan requirements of coaching or external support to crack CAT.
All the best.

Irrespective of the stream, IIMs are always better than every other institute in India.