Bihar

Based on your JEE Main score, you can get admission in any government college in any state.
But, it could also depend on the regional cutoff and availability of seats in a government college.

You need to check with the exam conducting authorities about this information but I guess you can use your permanent residence proof of Bihar. However, you can check this information from the exam helpline number for clarification as they are the right party to answer.

For better understanding, kindly mention for which purpose do you wish to present the documents.

See, if you clear those exams then you can easily appear for the paper but before that appearing for the paper is of no use.

Kindly name the college correctly as I am not able to fetch any college name with the one that you have provided.

Yes, you appear for IIT IN 2017.

Except Jadavpur University, each and every government engineering college considers a West Bengal domicile as a must.
Even though you have a Bihar domicile certificate, you can get a WB domicile, but in some cases. Visit the WBJEE site, you'll find some formats. Those are for domicile certificate required to be shown in Reporting Centres after counselling. You can get the certificate only if you are residing in West Bengal for more than 10 years or if your parent (s) are resident of Bengal.

I do not have any concrete information about Mahatma Gandhi University, Kolkata branch in Bihar or Patna. But, Mahatma Gandhi Central University is available in Bihar.
Please click the link below for more information about the university.
http://www.mgcub.ac.in/.

Yes, it is good. But the suggestion is to visit personally for getting the knowledge on the facilities and infrastructure.