Hi ! There is no course calling robotics to be very honest. It is an elective course and you can learn it from elsewhere. Its a non productive course too. In m.tech you can have some great other streams. Robotics is sort of a skill or project which will add to your CV. So hope you understand and best of luck to you.

yes ofcourse.

you can study mtech in robotics after studying btech in CSE.

but then you will have to prepare for gate accordingly with different subjects.

pls refer to the following link for more infomation:

https://engineering.careers360.com/exams/gate

if you want to study MS/MTECH from outside india, it will not be a problem.

hope this helps.

good luck