Hi Aditya. 

Your question reminded me of the similar kinds of problems and dilemmas I faced a few months back, during my preparation days. My academics are almost like yours, which proved quite advantageous, as I managed to get calls from good institutes even on a decent, not so great

IIM and top Colleges have maximum weightage on CAT scores and WAT PI/GDPI Scores. There is no such colkege focusing on academic.

So you need to brush up communication skills from Toastmaster International and be fluent enough as ther eis no escaping in terms kf spesking fluently if you are

IIM A has its own strategy of shortlisting candidates including their CAT results, they give you a chance of total 30 points to be earned. In case you get more than 90% in 10th, 12th, B.E., you will be awarded with 3 marks for each, if you get more than

Very good chances although your call would largely depend on CAT percentile.

If above 90 percentile in General category you can get majority of calls from top IIMs. 

Based on cutoff you can have follwing chances. 

https://bschool.careers360.com/articles/cat-cutoff

Hello Pranay,following are some good institutes:
Alliance University

Christ University, Bangalore.

Xavier Institute of management and Entrepreneurship.

The Bangalore University.

Hello dear, 

In order to take admission in IIM, you have to clear: 

1) common admission test( cat) 

2) after clearing cat, you have to give written ability test ( wat) and personal interview(pi). 

The student must have a bachelor's degree with a minimum of 50% for general category. So

For IIMs, necessary criteria for 2 year programs is that you need to get very high percentiles in Cat. But if you are a foreign national, Gmat is acceptable in your application. Instead if you have more than 5 years and preferably less than 12 years of workex, you can