Masters

Yes you are eligible for graphics designing.

That would depend on what graduation you have. They may accept it. It is hard to say without knowing.

Yes, they offer MA in communication
https://www.shiksha.com/mass-communication-media/course/m-a-in-communication-university-of-Hyderabad-gachibowli-82763

Master's/Direct Ph.D. For those students possessing a Bachelor's degree from an accredited college or university or its equivalent, should have a target undergraduate overall grade point average (GPA) exceeding 3.2 (A = 4.0). This 3.2 GPA requirement exceeds the university requirement of a 3.0 GPA. There may be other requirements. Contact the international students desk of the university for more details.

Both have good scope. But, it depends on your interest where you want to go.

It is all about weighing your options. There are pros and cons for whatever decision you take. Studying outside will give you global exposure, higher standard of living, good credentials, international experience etc. , but at a high cost. Studying in India is cheap and if you are really good at your studies and if you study at a top rated school in India, then you get almost everything that I said for studying abroad aswell. So weigh your options and then decide.

M.Sc Acoustics. M.Sc. Applied Electronics. M.Sc. Astronomy. M.Sc. Astrophysics. M.Sc. Electronics. M.Sc. Geophysics, and many more.

Yes, you are eligible. You can apply now.

Yes, you are eligible. But please double check with individual institution.

You can go ahead with the course, as it has great career and opportunities ahead.