GATE

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.

It's just from my own experience that I am saying. In Delhi, I am aware if one good institute which is Made Easy. It is good institute which covers all the topics and they have a very good teaching faculty members. They are even good at doubt clearing and moreover their notes are very good. I have seen their notes and it good. If you just practice that you will be confident enough to crack GATE.

Only 3 months left. That's very less time. Still don't lose hope. I am not sure what is your branch so I cant be specific. I can be general. So make your concepts clear in the first month and don't try to cover each and every subject. Just focus on the subject which is maximum marks and also the subject which are the easiest one. Just focus on these subjects and solve previous year papers as much as you can.

You can opt for B.Tech in any branch you are interested in. If you want to join core companies then go for Mech, civil, EC, EE. If you don't want to go for core companies and rather want to pursue your M.Tech, then it depends upon your interest in which field you are interested. If you are interested in maths and coding, you can go for CSE IT. These are the options you can always go for. You can start preparation mid 2nd year or in third year. But early is best. So, 2nd year will give you sufficient time to solve problems. All the best.

If you are in your 2nd year, then you can join any good coaching centre in you area which is a good one. If you don't want to join any coaching centre, then you should watch the NPTEL videos which are very good from GATE point of view. The most important thing is practice previous year papers as much time as you can. It will make you solving skills fast enough which is a major factor.

You can prepare for GATE either when you are in your 2nd year or in your 3rd year. You haven't specified the branch. So, I can't be specific. I can just say that practice from previous year papers and join any coaching institute of you can. If you can't it won't be a problem, if you have time Watch NPTEL lectures and you will be good to go. All the best. Just practice and practice.

Yes, you can appear for GATE more than twice. ECE us a good branch but their scope in India is less if you want a job after M.Tech or B.Tech. Usually, doing M.Tech in ECE is good than only B.Tech. Moreover after M.Tech many companies doors are automatically open for you. Also some PSU may give admission based on that and the score you achieved.
I hope it helps.

You can prepare for these branches in two ways:
1st you can join any institute like made easy or Ace. They are the good ones and they make the concepts clear. If you don't want to spend money in offline coaching then
2nd choice if you have ample amount of time you can always go for NPTEL and self Study. EC and EE are the most difficult branches from my 4 year of experience I am saying that. Always try to learn and practice previous year papers. Which will be very helpful.
I hope it helps.