NIT

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

You have got good marks, but NIT, Surthkal, Karnataka comes under the top list of NITs.
Not only this, they both are famous for their mechanical branch after NIT, Durgapur and Surat.
So, 139 is not enough as the admission in those NITs starts from 180+ for home quota and 250+ for other state students.
But as you mentioned you are an OBC category student, you have very slim chances to get into it.
I will prefer to give them top priorities in the choice filling.
All the best.

For getting into IIT, you must qualify JEE-Advance.

Try the below link of Shiksha:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor