IIM

Hello,
No if the sectional cutoff is not cleared you will not get call for GDPI process.

Hello,
There is no such provision.

Hello,
Yes, you can get admission.

Hello,
In IIMs, you will get all types of specialisation like Marketing, Finance, Operations, Analytics, HR, etc.

Hello,
You have good chance of getting admission into old IIMs.

Yes, you have that opportunity offered by IIM Indore. It offers a 5 year IPM (Integrated Programme in Management). IPM is aimed at students who have passed out of class XII/Higher Secondary or equivalent from schools across India. Please go though this for further details: https://www.iimidr.ac.in/academic-programmes/five-year-integrated-programme-in-management-ipm/
All the best!

You need to have 99+ percentile to get into top IIM. To know more about the same visit: https://www.shiksha.com/mba/articles/cat-2016-how-much-percentile-is-enough-to-aim-for-top-iims-blogId-13048.

Yes, you should appear for CAT exam and clear it too with a good score like not less than 98 or 99 to get into an IIM.

IIM Lucknow is better in terms of placements and education however every IIM is excellent in their term. So, decide as per your convenience too.

If you are getting the best B-Schools abroad like Harvard, then it's really great but in India, any IIM is best to study from.
All the best.