Bihar

Hello Raunak,
Please refer to this link to get more insights of the colleges and branches:
https://www.shiksha.com/resources/college-comparison
However, BIT-Patna is better than Heritage Institute of Technology.

HHello,
Greetings from Shiksha!
Check out the college predictor of Shiksha:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
You will find your college through this. Good luck!

See eligibility, admissions process, fees, course details, reviews and other details. please go through the link given below:
https://www.shiksha.com/bba/colleges/bba-colleges-patna

Hi Dear,
Kindly use the JEE Mains college predictor app to get more insights about the shortlisted colleges for your JEE Mains score. https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor

As of now, you don't need to think about problems. Just plan your area of interest and go ahead.

Yes, you can take admission in various Government College. Here are some links, please go through it.
https://www.shiksha.com/b-tech/course/b-tech-lateral-entry-government-college-of-engineering-amravati-gcoea-239643
https://www.shiksha.com/b-tech/course/b-tech-lateral-entry-institute-of-engineering-and-management-salt-lake-city-kolkata-235244

You need to obtain a domicile certificate for getting this benefit. You can go back to Bihar and apply if you need a state quota. To get domicile from Madhya Pradesh, you need following document and you may apply for the position of a Tehsildar or a citizen service centre:
The proof of the continuous resident of the state for a minimum period of 15 years.
Deed of Land in the state
Valid identification proof
School certificates
Copy of Ration/Voter ID card
Tehsil Enquiry report
Parents/Guardian domicile certificate which states that they are permanent resident of the state in case you are a minor.