NIT

Yes, they do. Check out the link below for details:
https://www.shiksha.com/science/exams/iit-jam

Many national and international firms visit the NITs, but there is a difference in firms visiting the top and average NITs.
You can check out the link below for details:
https://www.nitt.edu/home/students/facilitiesnservices/tp/comp/
http://nitp.ac.in/nitpold/placement_companies.php
http://www.mnit.ac.in/placement/

Share your JEE Mains rank and the engineering stream you want to opt for better assistance.

As far as I know, only NIT-Rourkela is offering a dual degree integrated course in Master of Technology. A dual degree from NIT is a good option, but it will bind you to the same institute depriving you of any better opportunity in future. It will be a far better option to pursue your B.Tech from NIT and then try for M.Tech from IITs.

What is your score?

Unfortunately, you can't. National Institute of Technology accepts only JEE Mains paper 2 rank to admit students in B.Arch. IITs accept NATA score. However, there are some other colleges which accept NATA score.
Hope it helped.

If you want to get into any NIT then NIT-Mizoram or Nagaland are the best options for you. They have home state quota.
Work hard and score good marks.

The cut-off for CSE in BITS-Goa lies between 317 to 330 marks in 2017. I don't know the exact iteration number, but probably it's 3rd or 4th.