BBA

No, not simultaneously unless it is an integrated course leading to MBA.

You should try for Christ University Bangalore ITM Mumbai, Balaji Institutes Pune, MITSOT, MITSOB, FLAME, NDIM Delhi AIMS Bangalore. These are some of the best Institutes which you can apply for with your MAT score

If you think you can score more than 70% in 12th after improvement then go for it. else dont waste time if again you would be stuck in 60s. as entrance exams can help you get admission with 57 too. But a few MBA colleges will not accept you if your score is less than 70

Doing regular course always have higher weighage. As you are going to pursue graduation my suggestion to you is go for the full time as when you will go for post graduation at that time you might be willing to pursue some course which might not accept your distance degree.

Hi Pragati,
If you want to do only MBA after graduation then BBA is a good option but only from good recognized college. But if you want more career options after graduation then b.com is good subject. You will have to keep in mind that BBA have more theoretical part while B.com have more calculations and practical part. Hope this answer helps.
Best of luck for you future.