Colleges in Uttar Pradesh

No, it's a private college.

Hey,
Yes, GBU provide the courses for foreign languages.
This may help you:-
https://www.shiksha.com/university/gautam-buddha-university-gbu-greater-noida-38229

The course is available in BHU. It is a full time course available in university. It is a 2 years course. The total fees is around 5000.

Ph.D eligibility criteria:
The minimum educational qualification required to apply for admission to the Ph.D programme is as follows:
1) Master's degree in Sciences/ Engineering/ Technology/ Management/ Social Sciences/ Commerce with a good academic record.
2) B.Tech /BE degree holders with a minimum of CGPA of 8.85 on a 10.0 point scale, or with a valid GATE Score.

yes, you definitely are as their eligibility criteria says:
The basic eligibility criterion for pursuing MA(Public Administration) degree is having graduation degree in any stream with a three-year study duration. Candidates should have a minimum aggregate of 50% or its equivalent CGPA.

No, I don't think so. You can only do it after passing class 12th.

Yes, admissions are open here. You can apply.
Good luck.

Yes, your rank is good enough to get admission here. You can apply for admission.
All the best.