CAT

Registration starts from August and ends in September for CAT 2019 and exam will held in November, but it's not possible to get admission this year so you have to wait for next year.

To get FMS, Delhi you should have 95 percentile at least and after CAT FMS take their own test. To get admission here you have to perform well in both CAT as well as FMS test.
Good luck.

Yes, after graduation you can do CAT. You can take CAT exam in your final year of graduation. You will be eligible for that. Good luck.

Here is the complete admission process to get into this university, you can follow the link:
https://www.shiksha.com/university/amity-university-Noida-41334/admission

You need to appear for the CUSAT examination.
You can get detailed information about it from this link:
https://www.shiksha.com/university/cusat-cochin-university-of-science-and-technology-kochi-28013/admission
Best of luck.

According to your rank, there is good chance to get this seat.

As per the eligibility criteria for MBA at Amity University, Noida, candidate should have passed with 50% marks in Graduation with valid scores of MAT/ GMAT/ CMAT, CAT/ XAT or NMAT.
Or,
Amity Written Test on the day of Interview.
So Amity does not prefer its BBA students for MBA programs.

I am afraid, one needs to have scored minimum 50% marks at graduation level to get admission into MBA. Since you have only 47% you don't qualify the minimum criteria for admission into MBA irrespective of the score you achieve in CAT.

No, this percentile is not good enough to get any IIM. Your percentile should be above 95 to get IIM. So, you have to prepare well. Good luck.

Doon Business School. They have placements in companies like Decathlon.