M.Tech

GATE is most probably required for M.Tech in civil engineering.

So far, I haven't seen a college which recruits on GRE in India. Admission happens depending upon on the score of GATE, or state PG exams. You can also try for management quota, but fee will vary from college to college.

The Master of Engineering (MEng or ME) are postgraduate Master's degrees offered by Canadian Universities that is achieved for higher studies in one of the engineering sciences branch or subject. Due to the numerous possibilities for specialisations in the area of Engineering Universities also offer degree types such as the Master of Science in Engineering (MS in Eng.). MS in Engineering subject is equivalent to MEng or ME or M.Tech ( awarded in India)

You are eligible to seek admission in IIT for M.Tech by appearing in GATE.

Check this link for all information regarding ME admissions:- http://bitsadmission.com/hdmain.aspx.

Please refer to the link provided below for details regarding admissions in various courses in Amity University. You should be able to navigate to a course of your choice awarded by Amity University. Decision to study a particular course is entirely yours, and if you feel that it will add value and will be useful for your future, you should definitely pursue a course with amity.
https://www.shiksha.com/university/amity-university-noida-41334

No, it's not possible. To pursue M.Tech, you need to have a B.Tech degree.