DTU - Delhi Technological University

Yes You are eligible. As DTU eligibility Says :-
60% marks in 10+2 in 10+2 marks in aggregate of Physics Chemistry and Mathematics along with English as one of the main subjects from a recognised board. Also the admission are done through entrance examination also known as Joint Entrance Examination {JEE}
and on the scores basis you can do your admission in the same. Well hope You get your answer.

For getting admission and a sure seat in DTU, your JEE Mains general category rank must be under 10000. That means you have to score at least above 99.2 percentile in JEE Mains. For top branches in DTU, your JEE Mains rank must be under 2800 rank, or you can say more than 99.8 percentile score.

Yes, if you have already done with UG degree then you are eligible to appear for GATE exam, keep checking up with the official site of the same where they release forms to appear in the GATE exam.

That is not possible. Even lateral entry option I is for second year.

Delhi Technological University (DTU) conducts the Lateral Entry Entrance Test (LEET) for admission into its lateral entry programs. DTU LEET is the abbreviated form of the Delhi Technological University Lateral Entry Engineering Test.

Chances of getting DTU for B.Tech industrial and production engineering are moderate. However, the chances can be made for the following.
1. Environmental Engineering (ENE)
2. Biotechnology (BT). .

There are lots of options for Science students like B.Sc. BBA B.Com agriculture and other. If you like mathematics then you can choose B.Sc. In mathematics.