Exam

Yes, you are eligible.

Your rank is not so good. So, there are very less chances to get this branch.

You can see the opening & closing rank by Shiksha link:
https://www.shiksha.com/hospitality-travel/hotel-hospitality-management/exams/nchmct-jee/cutoff.

Score above 160 and you'll be able to get a seat in most of the top branches. If your concern is only about placements, then I would suggest go for Computer Science (CSE), IT or Computer and communication (CCE). Just keep your CGPA above the minimum eligibility criteria and you are good to go! Howev

Yes. They accept MAT.

It will depend upon the NIT reputation or how old is that NIT is. By the way, don't take tension. You will definitely get the college of government only. You can get the MNNIT Allahabad, NIT Patna, etc. like this, you get at the counselling.
Best of luck.