Engineering Entrance Exam

No you are not eligible for admission in NITs and IITs. The eligibility criteria is given below. Eligibility : 12th with minimum 75% of marks if belongs to General and OBCs, while 65% for SC/ST. Or should be in top 20 percentile of respective board.

Hello Jaideep, I am afraid that in Pune to get admission in an engineering college you should have appeared in the entrance exam conducted by the state of Maharashtra called MHT CET. Ranking in JEE Mains cannot get you any seat in the government college. However there are deemed to be universities and private colleges which may give you admission based on ranking in JEE Mains. I am attaching a link that will give you all the required information about different engineering colleges in Pune and all related information. https://www.shiksha.com/b-tech/ranking/top-engineering-colleges-in-Pune/44-2-0-174-0
Best wishes to you.

Yes, you will get lateral entry in college for degree. Year gap is not an issue but you can only apply GATE exam after completion or in final year of your engineering degree.