M.A

Hi,
If you focus and goal is set to enter into media industry than you can go for BA journalism, otherwise, you can do BA English Honors and then enter into MA level through which you can improve your communication skills and in 3 years you can do internship in the same to understand in which profile you would want to start your career in journalism, whether it's English journalism, reporter, news anchor, research, content writing or in production line.
So, decide according to your interest.

Yes, you can do MBA after a gap of 3 years but if you were working it can come as a plus point for you as experience.

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.

You have not mentioned the subject in which you want to do MA. You may reframe your question for an expert to share advice.

A postgraduate diploma is a shorter qualification than a masters degree, although at the same academic level. Whereas a masters degree is made up of 180 credits (taken from exams, assignments, and so on), a postgraduate diploma is 120 credits. There is a shorter version of this too a postgraduate certificate, which is 60 credits. There are a variety of courses that come under the heading of a postgraduate diploma it can be a vocational course, such as a legal practice course, or an academic course.

The university has 3rd ranking for MA economics, so this would be a safe choice.