Engineering Entrance Exam

Yes, both B.E. and B.Tech Candidate can apply for GATE exam as there is no difference between BE and B.Tech degree just a different name for engineering degree.

Hello Siddharth,
You may use the college predictor for JEE Mains 2017, which may suggest some colleges that you can get:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
The link below gives you the previous year's closing and opening ranks for each NIT, IIT, IIIT, etc.:
http://josaa.nic.in/Result2016/result/OpeningClosingRank.aspx

No, it's electrical in CUSAT because it is going to become IIEST this year and IEM stands nowhere.

Little chance it is risky as your rank is not too good so if you are satisfied with you branch then you can join the college and right now lock the college for future reference.
All The Best

That's good you get good NITs college as your rank is too good. Just check the college which you can get in the link provided:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

Go with the JEE Mains counselling and you should search for some good private institutions near your region. It would be well and good if you consider Narsee also as an option.

You would be asked to fill the colleges and choose the preference. You have to check the details too for a better verification of your admission. results would be announced after few days.

First of all, I want to suggest you that please make sure before preparing for JEE Mains that you really want to do engineering, ask yourself whether it really interest you or you are doing, because everyone is saying to do so.
Yes, if you really want to do engineering, then I suggest you to immediately take admission in any coaching classes.
Or, you can also do one thing, just do your class 11th and class 12th thoroughly, and give IPU CET. If you will score well, then you can get a good college to be in.
If you have done your class 11th and class 12th thoroughly, then it is easy to crack IPU CET with good rank.

Yes, in all NITs, national level institutes or central funded institutes admissions are done on the basis of JEE mains merit ranking.