Engineering

Yes, you need to make payment.

You can contact the authority:
https://jeemain.nic.in/WebInfo/Public/View.aspx?page=78

There is a state-level entrance examination for lateral entry which will give you the colleges according to your rank in the respective states. Like LEET, PTU LEET, LPU NEAT Calicut University Lateral Entry.

NIT-Mizoram is the best option for you at this score.

It depends on other students performances, but according to the trend, it should be above 9.

Scope of metallurgy is good whether it is in core or non core field. Non core field has already mentioned in other answers like you can be a business analyst, can go in consultancy companies etc. You can do MBA, IAS, MS, Mtch, MSc, Mphil etc. Many non core companies come for metallurgy as well in PE

You have chances to get because they have reduced the shift, not seats.

http://aerospace.gateing.com/gate-aerospace-cut-off/
you can see this.

If you want to pursue MS from a really reputed university like (in US) MIT, Stanford, Harvard or Wharton, then you have to pay a minimum of one crore rupees.

Please checkout the given link:
https://www.shiksha.com/university/sharda-university/exams/Surat
Some good books you can refer for preparation:
Concepts of Physics (Set of 2 Volume)H C Verma
Problems In General Physics I E Irodov
Handbook of PhysicsNipendra Bhavnagar
Chapterwise Topicwise Solved Papers Ma