Babasaheb Bhimrao Ambedkar University (BBAU), Lucknow

Hi,
There is no such option available where you have to choose any specific subject. You can still get in touch with the college officials for detailed information at:
Toll-Free: 1800-180-5789
http://www.bbau.ac.in/new/index.aspx

Yes. you can apply for this.

Here is a link with some helpful information for you, http://recruitmentresult.com/bbau-entrance-exam/. If you need professional help in preparing for Ph.D. entrance exams, visit http://shelldreamsoverseas.com/. They are best Ph.D. consultants in India and can help you get maximum benefits from studying Ph.D. abroad like 100% scholarship, handsome stipend while studying and also tuition fees waver. You can call them on Ph : +91 79 4006 7258 M : +91 99980 23693 +91 99980 23694.

Admissions are closed now. The forms are made available in the month of March.

Both Hindi & English medium.

Sorry, M.Sc is not offered in BBAU. To know about the courses offered, refer the link: https://www.shiksha.com/university/babasaheb-bhimrao-ambedkar-university-amethi-satellite-campus-lucknow-51351/courses

First, you have to graduate and yes, you are eligible after the completion of your graduation because graduation in any subject can apply for Master's degree in library science.
Good luck.