History

Yes, it is recognised for all the purposes.

Yes, You can do it after engineering.
Through the distance mode: IGNOU
If you want to do in some prestigious institutions:
By joining in college: JNU in SSS(School of Social Sciences) or SIS (School of International Studies). The latter will focus more on IR.
There are many other good institutes like DU, Central Universities soon. For both of these after applying you have to write an entrance test.

Yes, graduation is necessary for any professional course.

JJT University does not offer PG courses in Ancient Indian Culture or History. You may refer to following link for courses offered by JJT University:
http://jjtu.ac.in.

Hello,
Understand that the admission decision is up to the sole discretion of the college. The admission decision will depend on your overall profile and the selectivity of the college. So, nothing can be predicted. You should by the way specify more details such as your academic grades and also what you want to study for detailed advice.