Engineering Entrance Exam

You can apply for both under one counseling.

You can also get in government colleges.

Yes, your friend can get a good NIT.

It is combined but individual cut-off also comes as there is individual cut-off for chemistry, maths and physics.

Unfortunately no. All admissions and entrance exams happen much in advance even for a private university.

Hi,
Try the college predictor:
https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor&ved=0ahUKEwiemPTz9eXTAhXMRo8KHUM4DiEQFggbMAA&usg=AFQjCNEdcXmQi2OdKUdFdfXRk0v1xEEY5w

It seems quite difficult but still, try the college predictor:
https://www.google.co.in/url?sa=t&source=web&rct=j&url=https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor&ved=0ahUKEwiemPTz9eXTAhXMRo8KHUM4DiEQFggbMAA&usg=AFQjCNEdcXmQi2OdKUdFdfXRk0v1xEEY5w

Yes, but in good colleges. It is better to try in private universities or you can also try next year.

The Graduate Aptitude Test in Engineering (GATE) is an examination that primarily tests the comprehensive understanding of various undergraduate subjects in engineering and science. The score is used for admissions to various post-graduate education programs (e.g. Master of Engineering, Master of Technology and Doctor of Philosophy) in Indian higher education institutes, with financial assistance provided by MHRD and other government agencies.