GATE

For general candidates, NITIE cutoff is 720 - 750 + GATE scores. Fee is 3.7 Lakh rupees per year.

Hi,
GATE provide you college for PG not for Ph.D. For Ph.D, you should NET or entrance exam for admission.

H8, please refer to the link provided for details:
https://theprint.in/india/education/from-next-year-nit-students-to-get-direct-admission-for-phd-in-iit-Delhi/314705/?amp#aoh=15758176644388&referrer=https://www.google.com&_tf=From %1$s.

Yes, you can give GATE exam through physics.

You should make certificate of it and apply under pwd quota as your right ear is not working properly. If you apply under general or other quota, you might get rejected during medical as they conduct ENT test. So, better to apply under pwd.

You should score 50percent of the total marks. Best of luck.

Hi,
Here is information about GATE
The Graduate Aptitude Test in Engineering GATE is an examination that primarily tests the comprehensive understanding of various undergraduate. Score/grade validity: 3 years GATE 2015 onward)
Annual number of test takers: 718854 (in 2018). Year started: 1984
Qualification rate: 16.53 % (in 2016).

Yes, there are some colleges which recruiters through GATE exam. But, government college not.

I have no information regarding which nits are selected for this programme by IIT Delhi. But, there are some points you should focus on :
1. Students will be required to spend a minimum of six months or complete a project with the IIT during their last year of B.Tech before they can start with their Ph.D. programmes.
2. NIT students, who have a CGPA of 8 and above in their third year and are interested in research will be able to get direct admission in IIT Delhi for completing their final year here. After that, they can continue to pursue their Ph.D. in the institute.