CAT

Yes, it is good but you have to appear for CAT and the percentile you get there will act as a deciding factor along with these to get into IIMs. Get more than 95 percentile to get into a top IIM.

Yes, surely, you can get admission in ABBS. Just go through ABBS college website and apply for the admission.

Your chances of getting a call are very less but still you can try.
All the best!

50 percentile is the minimum cutoff for for all national level management exams. You better try once more.

Do not join any college even if you get admission. It will be better to try again.

Yes, you definitely have a good chance of getting in IBS Mumbai. As far as your marks are concerned they are very good. The only problem you might face at the time of placements is that some of the companies have an eligibility criteria of 60% and above in SSC, HSC and Graduation. Rest is all good as only few FMCG companies and some of the banks have this criteria. You are 100% fit and eligible for MBA and getting good placements by looking at the academic records .

It depends on your caste and category. If you are from general category it is still hard to get a good IIM with 95.41 percentile.

78.19 percentile is a good score but the trend of TAPMI is to take percentile above 80. You can apply with XAT separately. How much did you score in XAT?