CAT

IISWBM does accept MAT scores, only CAT scores are entertained.
A CAT score of 75-80%ile may fetch you a call for GDPI, after that your profile academics, work experience, may take into account.
GDPI call are done solely on CAT %ile basis and no profile based calls are done.

75-80 percentile in CAT may fetch you a call for GDPI.

Apart from IIMs, you can get calls from top B-schools like NMIMS, SP Jain, MIT-Gurgaon, IITs like IIT-D/B/KGP, JBIMS, SIBM, SCMHRD, XLRI, XIMB and IMT. You can check the below link for top MBA colleges along with fees and placements information:
https://www.shiksha.com/mba/ranking/top-mba-colleges-in-india/2-2-0-0-0

Well that will depend upon your CAT percentile. You should have atleast 95% in CAT if you are thinking for IIMs. Secondly, your graduation score is still less but on the other hand it's diverse (Physics Hons.), which is plus point. So, give it a try and all the best!

Some of the colleges you may look for-
Amrut Mody School of Management, Ahmedabad
PSG Institute of Management, Coimbatore
Jaipuria Institute of Management, Lucknow
School of Communication and Management Studies, Cochin
Kirloskar Institute of Advanced Management Studies, Pune
Thiagarajar School of Management, Madurai
Siva Sivani Institute of Management, Secunderabad
AIMS Bangalore
Prestige Institute of Management, Gwalior
Pune University Dept. Of Management, Pune
Lovely Professional University
IMT Nagpur/Hyderabad
VJITM, Hyderabad.

There are many entrance exams for MBA in India. Only CAT exam is not necessary but it's mandatory exam if you want to join in top colleges and Iims. And it's one of the toughest exams in India.

I don't think so that Sri Balaji Society gives admission based on category that is General, SC, ST, etc.
The cut-off for CAT in general category is somewhere around 80-88 percentile.
But, one thing I can suggest you is, if possible fill the application form, and if you get a call from SBS, try to perform well in GD/PI, because they give more weightage to GD/PI.

Yes, you can get admission in FMS as you have a good score.
Best of luck.

There will be certain questions asked about your graduation score but if you have scored well in MBA entrance exams then I don't think there would be any problem.