NIT

No, you can't take admission in NITs. NIT need75% marks in class 12th board for admission.

Yes, you will get the admission in IIIT Hyderabad.

Yes, you can give improvement exam for any subject of 12th class for meeting the JEE Main eligibility criteria of 75% aggregate marks. The CBSE improvement exams form will be available in the month of October every year.

In JEE there is no weightage for board marks everything depends upon JEE Main percentile. With 82 percentile you mayn't get god NIT's you get only north side NIT's.

Your expected JEE Main AIR is about 63649. You have not write about the category you belongs. However keeping the last year of trends in consideration and if you are from general category with Odisha home state, than you can get the following engineering branches.
1. Food Process Engineering
Further if interested in Integrated M.Sc. than you can get the Integrated M.Sc. Chemistry.

No actually you are not eligible for admission in any government college, NIT, IIT or any other reputed private college like VIT, SRM, BITS PILLANI etc. This is because the eligibility for these are minimum 75% in class 12th. But definitely you can apply for any other private college those who don't consider this eligibility criteria. ALL THE BEST.