B.A

Every course is worth if only you know what to do with it and why are you pursuing it. So identify your choices, strength and skills. Look at the course structure of courses like B.A history, sociology, philosophy, geography, anthropology, psychology etc and see which one is of your interest. Plenty of options in research, education, administration, banking, ngos, CSR are available.

If you had mathematics in class 12th, you can do MCA after BA.

It is recommended that you do either MA Economics or MBA Finance to have a good career. You chose any based on your interests to excel.

Yes, you can get admission in IIM after clearing CAT/ GMAT and getting a good score in it.
To appear for CAT etc  student needs to have 50% in graduation. But eligibility criteria varies for various Management Institutes or Universities, they can take different %age in graduation along with these test. 
These days we have many other B-Schools that are at par with IIM. Many of them offer admission on basis of these tests/ their own tests/ GDPI/ marks in previous qualification.
You can check the websites of universities like Amity, Lovely, Chitkara, Manipal etc for more details.
Good luck.

It is one of the best option either. If you go through history of successful candidates, you will find a good number of them from the field of Arts. Don't think much. Go ahead and pursue your goal. You will surely get success. God bless you.