Lateral entry

Yes, it is possible for lateral entry but it depends on college to college and university to university.

Yes, you only have to clear GATE or any government services exam.

Try Thapar Institute of Engineering & Technology. They have BTech lateral entry in Chemical engineering, Civil Engineering, Computer Engineering, Mechanical Engineering, Electrical Engineering, Electronics & Communication Engineering. Their 2018 admissions are on, you can check their website wwww.thapar.edu for more information.

It depends on your JEE Mains and advance score, and also from where you do your B.Tech, but let me give you an advice please be more specific, while you ask questions, because it becomes very difficult to give correct answers. And, wrong answer will do wrong to you only. I'll suggest you to go for mechanical or IT.

Yes, you will be eligible for the LEET exam, your aggregate percentage should be above 60%, even if you are not able to get 60% in one year, it doesn't matter.