IIT

Mechanical Engineering is one of the oldest branches of engineering with an extensive application platform that ranges from design and development to construction of physical devices. In 2013, the intake for mechanical engineering was 1,125 out of total 9,867 seats were available with IITs and ISM Dhanbad.

If you think you can work hard, then you should. If not, then go for a local college.

Yes, you can give Mains 3 times and Advanced 2 times.

Yes, you can take admission into M.Tech course without GATE score card. You can take admission in state technical university affiliated colleges by appearing for an entrance exam conducted every year by respective state university/colleges etc.

Clear GATE exam, apply for the college/PSU if you clear the cutoff you will receive call for the same.

Yes, it is possible, if you work hard and get a decent score in GATE. I think you can wait and you may get a chance in any of the good NITs.

Deeply clarify what actually you want to make your career. If it's maths, go for M.Tech. It's a higher qualification than any M.Sc. It gives easy placement and better job prospects.

Yes, you can take admission in IITs. You have to qualify JEE Advance in order get admission there.

Hello,
There are a lot of colleges to choose from. You can refer the below link designed by Shiksha, in which college ranking, their ranking, facilities all criteria are shown, follow the link:
https://www.shiksha.com/b-tech/ranking/top-engineering-colleges-in-india/44-2-0-0-0.

That totally depends on the pattern and difficulty of the paper and also the number of students applying for JEE. Percentile is calculated wrt to the number of students getting marks below the respective person whom percentile is being calculated. If the students will vary they percentile may also vary. I hope you understood.