Science

B.Sc in Exercise and Sports Science- Manipal School of Allied Health Sciences
B.Sc. In Physical Education, Health Education and Sports - LPU - Lovely Professional University
There are a few popular sports courses colleges in India. Check here- https://www.shiksha.com/business-management-studies/sports

You can refer to the entrance prep guides that you can download from here https://www.shiksha.com/science/fri-entrance-exam-question-papers.

Yes, why not. But sometimes, it also depends which course and college you choose for your study. There are a lot of very reputed colleges in Haryana and they provide good placements. Manav Rachna International Institute of Research and Studies
KR Mangalam University, Gurgaon
Kurukshetra University
MDU

Once completing M.Sc in Zoology, you can consider becoming a lecturer in College, or join some professional jobs in Government or business organisations dealing with animal and animal products like fisheries, etc.

Hostel life at IITI is one of the best in all the IITs. Newly constructed hostel building in the permanent campus has a great infrastructure.
Facilities are also great there.

You can go through the website of Jamia i.e jmi.ac.in and get the syllabus and then prepare accordingly.

Yes, you can apply for admission.

You can refer this link to understand the process in details: https://www.shiksha.com/exams/ntse.