Engineering

Refer the following link:
https://www.shiksha.com/b-tech/resources/wbjee-college-predictor

You might have a chance if you have any reservation criteria else give it a try. You have very great chance to get into this institute with this rank. Attend the counselling and choose your options appropriately. Try Shiksha college predictor for better analysis.

Yes, you can get admission only if you pass your maths exam because no college gives admission to failed students. So, first you have to pass your exam and with that result you can get admission in any private college. There is no chance to get a government college. So, take admission in a private c

Considering you're from general category, for mechanical you have slim chance though you may get some non-core branches.

Hello,
If you belong to general and OBC category then there are no chances for NITs but your marks are decent for getting good private colleges in your home state.

Yes, you can change the branch in second year but for that you need to top the entire batch and branch should also have a vacant seat, usually there are few seats vacant.