B.A Political Science

https://www.shiksha.com/humanities-social-sciences/political-science/colleges/colleges-delhi
1. Delhi College of Arts & Commerce:
2. Gargi College
3. Hindu College
4. Indraprastha College for Women
5. Jesus & Mary College
6. Lady Shri Ram College
7. Miranda House
8. Ramjas College:
9. Sri Venkateswara College
10. Zakir Husain Delhi College.

Yes, St. Xavier's College, Kolkata does offer B.A. (Hons. ). In Political Science. For further details regarding this course, please refer to the link provided below:
https://www.shiksha.com/humanities-social-sciences/political-science/course/b-a-hons-in-political-science-st-xavier-s-college-Kolkata-park-street-83765.

Delhi University, colleges under Calcutta University, etc. are the best for B.A. Political Science Hons.
Other colleges are also there-
Please check this link once.
https://www.shiksha.com/humanities-social-sciences/political-science/colleges/b-a-colleges-india

Yes, DU offers Army quota.

Yes, with 82% marks in boards, you are eligible for admission.

You can go for Ambedkar University or Delhi University. But, in 1st cut off I think it may be difficult. http://www.du.ac.in/du/index.php?page=cut-off-2017
You may check the upper link for get some idea of cut off (2017). But, there is variations in every course and every college cut off.

Hi Amana, Shaheed Bhagat Singh College has Morning and Evening Sessions. I am assuming you mean to take admission in the Morning session. Yes, you are eligible to pursue B.A. Political Science from SBSC if you are from SC/ST or OBC category. If you are from general category, you have chances to get admission in the second or third cut-off this year. All the best.

Yes. If it is possible for you to drop, then you can because it is better to pursue a course in which you are interested rather than doing some other course.

Hi Sake, Hindu, Hansraj and Ramjas, in the order of naming. They all are a part of the amazing North Campus. This brings you not only excellent campus life, but also some superb cafeterias. I didn't have a pretty nice experience with people at Hindu, but in spite of that, my vote goes to it. All the best.