M.Tech

Hello Karan,
Yes, droppers are also eligible for GATE exam as this exam is both for job and for higher studies, so you have to score good to get selected.
All the best.

You can go BITS, Pilani, VIT-Vellore, NIT and IIT through GATE.

You have a lot of options left with you as you have done you Masters. If you are very much interested in academics, then Ph.D is the obvious choice for you after completion of your M.Tech.
If not, then there are number of corporate giants that seek M.Tech pass outs as they want to hire subject matter

First of all, it does not matter how long you have taken to complete your B.Tech and IIT is a great choice if you pursue M.Tech from the same.

There are a lot of entrance exams for those who are going abroad.The exams are:
1. GRE
2. IELTS
After satisfying the eligibility criteria you have to show bank balance then only you are eligible.
Along with the bank balance, you must have a passport.
Instead of abroad, India also has good universities

You need to take GRE and IELTS or TOEFL.