GATE

You can get in many private engineering colleges which have their own set of criteria for admission. I would suggest you for UEM Kolkata and to know more about the college visit here:
https://www.shiksha.com/university/university-of-engineering-and-management-kolkata-47398

Yes, you can but you need to clear your backlogs before the completion of 4th year of your BE degree.

1- GATE syllabus is same as that of engineering subjects.
2- Revise all the core subjects.
3- You may join coaching depending on your grip to the subjects.
4- Start solving questions from a good book for GATE and get started with the pattern.

Yes, you're eligible for GATE. As long as your age is below 25, you also get to apply for the various PSU's via GATE. If not, then you can use the GATE score elsewhere to your advantage.

Hello Radhika,
GATE (Graduate Aptitude test in Engineering) is for admission into post graduate programmes like M.Tech
As the name implies, the questions are from the concepts that you would cover in the 4 years of B.Tech. For each branch of engineering, there are separate question papers so that you can do M.Tech from that branch itself.
In addition to this, you also have questions from general aptitude (GA).
Hope this helps you.

Yes, after your B.Tech, you are eligible for GATE exam as this exam is for higher study in IIT and NIT. So, you have to study well because the level of questions is high in GATE.
All the best.

Yes, why not? Go for it. I don't think there is any problem.
Hope it helped.