Hyderabad

I will suggest you that IMT Hyderabad is better than Nirma in terms of placement, faculty guidelines, infrastructure, etc.

If you are considering joining college without writing any exam. Than either you are looking forward to back door entry or low grade college because than only your wish will get fulfilled. Its better to give some entrance exam you can consider giving GMAT apply to some good colleges. Take your decision wisely its about your career.
All the best!

Yes, but you have to at least 70 percent to get this college.

Yes,  B.Sc in food and nutrition is available in Osmania University, Hyderabad. You can take admission there. Good luck.

Check this link for B.Sc courses that are available in Osmania University:
https://www.shiksha.com/university/ou-osmania-university-hyderabad-25100/courses/bsc-bc.

BIMM, Pune is better in terms of ROI and here almost 300+ companies visit almost all the sectors. The highest package this year is 14 LPA. IMT is a brand but IMT Ghaziabad is in Hyderabad with an average package of 8 LPA and minimum fees of 12 lakhs. The overall fees of BIMM is approximately 9 LPA. If you are getting an average package of 6.75 LPA then it is good.

All the colleges were start working by the end of forest counseling itself. You have to report and join immediately after the 3rd counseling, in case you get a seat, because the class work had already begun.