Colleges in Howrah

You can check which college you will get here:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

You must go for IIEST Shibpur CS, as it is one of the best in this country with great placement offers. Recently, Microsoft recruited some with INR 40 LPA. Without any doubt, I can assure you to join IIEST to have a great future ahead.

All the students must have passed higher secondary school (10+2 level) from a recognized university or council. A student can appear for JEE Mains only thrice.
The students must have also taken five subjects in Class 12 or equivalent qualifying exam. The five subjects should include Physics, Mathematics, one language, Chemistry/Biotechnology/Biology, and any other fifth subject. Students whose birthdate falls on or after October 1, 1992, will be able to appear for JEE Mains.
For JEE Mains Paper 2, a student must have Mathematics as one of the five compulsory subjects in Higher Secondary Examination.

Among the NITs, NIT Trichy is the best, academic wise then comes IIEST.

For general category, it is 60%. For reserved, there is relaxation of 5%. For more details check the official website.

Yes ofcourse you are eligible for GATE. This is one of the exam held for doing the post graduation courses in IITs and top colleges.

No, you can even submit your IIT-JEE Part 2 scores other than NATA with eligible 10+2 scores.

See, honestly, both the departments have good professors and core companies for your placement and to add to this look at the seat allotment after the 1st round. EE department has started from 29 rank and ETC dept has started admission from 51 rank, both are impressive.
So, to sum up, choose what you like. If you like working with a breadboard, resistor, capacitor, chips, etc., choose ETC. If you like to learn about motors, machines, transformers, etc., choose EE.
Good luck.