TANCET

If you have completed your graduation, you can. The eligibility of MBA is merely graduation.
You shall appear in entrance tests or exams on the basis of graduation. You need to do well in the entrance exams and that will take you to MBA.
The history you mentioned needs to be overshadowed by your good work in the selection test.

Depends upon your caste status. Actually, 37.33 is not that good score if you are from a BC. If you have luck, you will get Measi Institute of Management, I guess. It depends on your branch.

It depends on your community. Still, even SC members (the community which requires least marks) must have scored 50 or more for getting into PSG. PSG is the first college in which the seats will be filled during counselling.
Good luck.