IIM

Yes, you can study in any IIMS through CAT score.

Yes, CAT scholarship are available for students belonging to economically disadvantaged families in IIMs.

Call from IIM will depend on your score that has to be above 99% at least. Your class 10th and class 12th percentage are fine.

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.