M.Tech

http://www.modyuniversity.ac.in.

For seat in M.Tech, either you need to write state level competitive exams or take GATE exam at all India Level. Based on the rankings, you'll be getting seats in respective colleges.

The Minimum eligibility for Ph.D. is completing a post graduate and most of the institutes have entrance exams. You cannot get admission in any of the Institutes without completing PG.

The minimum required score for GATE is 650.

Yes, as for joining in M.Tech integrated course, you need to not attempt any entrance exam and the selection is based on class 12th marks. You can easily get admission in the course with this percentage.

VIT doesn't accept other entrance exams like JEE Mains. And VITEEE exam is only for getting admission in B.Tech programmes. You need not write any entrance for getting admission in M.Tech integrated course. The selection is based on 12th marks.

First of all, there's nothing like distance mode for engineering education. There at BITS, you have WILPs and for that, visit with this link: https://bits-pilani-wilp.ac.in/programmes-for-individuals.php
I would like to suggest that you inquire for recognition.
Regards.

No, you need not attempt VITEEE for getting admission in integrated M.Tech course. The selection is based on your 12th marks.

Yes, he can get admission in IIT-Kharagpur.
Kindly visit this link:
https://www.shiksha.com/engineering/course/m-tech-in-aquaculture-engineering-indian-institute-of-technology-kharagpur-293511

Yes, you have a chance of getting seat in AU as you got very good rank. So, attend the counselling conducted by the APSCHE and submit your details.
All the best.