Karnataka

Christ, Alliance, XIME and Ramaiah accept MAT scores. You can appear for MAT exam and apply to these colleges. Try to secure above 650 composite score to get admission.

University or college change completely depends on the university or college, some universities allow and some won't plus your first year marks should be good for the same, contact the concerned authorities for detailed information.

I would suggest you to join TIME or IMS as they have a pan India presence and have a good question bank of mock papers and help you with the preparation for GDPI process also.

You need atleast 60% marks in 12th to get into Christ.

Hi,
Top colleges offering BBA in India are:-
Christ University, Bangalore
IIM Indore (BBA+MBA)
Symbiosis College for Management, Pune
Amity International Business School, Noida

Yes, for XAT, CMAT, MAT February, etc. there will be a second round of application.

Yes, you can. But, your candidature would be considered under All India Seat. So, you would need a higher score in any exam that you choose to apply through.

ISBR is much better and has good placements assistance as of now, with decent opportunities for all streams.
https://www.shiksha.com/mba/course/master-business-administration-isbr-business-school-bangalore/109872