GATE

SRM University, Sonipat conducts its own exam for Ph.D admission. Ph.D Mechanical Engineering (ME)
Eligibility: Candidates must have passed Post graduation in the relevant discipline from a UGC-recognized University with 50% marks in aggregate. However, an applicant with M.Phil/CSIR-UGC/JRF/NET/SLET cleared in the relevant subject is exempted from the Entrance test. As per UGC notification, the minimum duration for Ph.D. is 3 years including course work and a maximum of 6 years (full-time/part-time).

No, you first qualify GATE then you can take admission in ME or M.Tech in reputed institutes. But there are some low private colleges to which you can apply for ME. But the thing is you will not get a stipend.
Good luck.

You can go for the exams conducted by BARC and ISRO which are usually conducted after the GATE exam.

No, you cannot participate in the GATE counselling. You need to be in the last year of B.Tech or have completed B.Tech to participate in GATE counselling.

There are some Navratna PSUs which can accept. But it is rare.
Good luck.

GATE exam is required to do M.Tech and also to get into some PSU (government-owned corporations are known as Public Sector Undertakings (PSUs) in India).

There is no college by the name, GATE.

Yes, it totally depends on you if you want to prepare or not. There are several extra opportunities inside and outside VIT for GATE preparation. You will get a platform to improve yourself and prepare for GATE level.

You can apply for M.Tech in private college and someone low-grade government colleges.