Colleges in Howrah

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.

Yes, you are eligible to apply for B.A(Hons.).