CAT Eligibility

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.

For CAT you need to have a full time Bachelor's degree. So, I think you should confirm once with CAT authorities. The contact you can get the official website of CAT. They resolve these on a case to case basis.

You should target at least 99, as it is highly competitive these days.

Apart from CAT and CMAT there are other college specific exams like SNAP, NMAT, etc. You may give it a shot. Note - all exams for admission in this year are over. Try for next year.

No, CAT is mandatory for admission into IIMs.

You passed 10. So, you are eligible for giving 12th exams also.

Please check for your eligibility to pursue M.Com from the criteria mentioned below:
Master of Commerce M.Com Eligibility:
For admission to M.Com. Programme, students must fulfil the below-mentioned eligibility criteria. Bachelors degree in Commerce
Minimum 50% marks at the level of graduation
Certain universities also admit students with a B.A. Economics (Hons. ). Degree
Students who have passed B. S. B.A. , B. F. I. A. and B.E. With at least 60% marks can also apply
Regards.

Hi Akshat,
Yes, you are eligible to appear for CAT exam. You need to register yourself on the official website given below:
https://www.aicte-cmat.in/College/Index_New.aspx
All the best.

Your chances of getting a call from IIMs are not that good, but if you score good marks like more than 99 percentile then you can get a call but in PI, you need to give a valid reason regarding your gap of 2 years.

It is going to be very difficult for you to get an admission in IIM Bangalore, without some good Work Ex and CAT marks, as IIMB gives graduation marks very high weightage in comparison to other colleges. And, after getting in you will actually need work extra hard to justify the lower marks in graduation that you have (but, with hardwork that is possible.)
Anyway, you can also look at IIM C and L which don't give as much emphasis to past acads.
Good luck.