NIT

NIT is the best option.

No, you can't get admission there unless you belong to SC or ST.

Hi,
The fee structure of NITs is very low as it is under central government.
It could be around 40000-60000 per annum but if you have scored more then try to get good NITs.

You can easily get into NITs with this score.

5740 rank is in ST category if we convert it into general rank it would be around 400000-500000.
So, at this rank, it is too difficult for you to grab a NIT in other state.
Try for NIT in your home state quota I.e. NIT Hamirpur.
all the best

Yeah, you can. If you score good in JEE Mains and advanced, then you definitely got NIT or other government colleges and for the rank, you can check here: https://www.resonance.ac.in/answer-key-solutions/JEE-Main/2017/jee-main-2017-rank-predictor.php

You have great chances to get NITs if this is your All India Rank. The cut off scores based on caste decides your chances to get the top NITs. If you have scored way more than your cut off then you can get good NITs.
The caste wise cut off is:
General (OC)-81
SC-32
ST-27
PWD-1
OBC-NCL-49
You can apply for counselling through www.josaa.nic.in
You are qualified to appear for JEE Advanced.

The counselling will start only after the JEE Advanced examination.
The date for JEE Advanced will be at the end of May. So you have to wait till that.
May be the counselling for JEE will be from second week of June only.
Apply for counselling through www.josaa.nic.in

Hopefully, this will help you:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

You can predict your rank and college with the help of the links given below:
1. Rank predictor: https://www.shiksha.com/b-tech/resources/jee-main-rank-predictor
2. College predictor: https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor