BITS Pilani - Birla Institute of Technology and Science

You can opt for electrical and electronics engineering or electrical and instrumentation.

Also,if you got under 100 rank you can and get 100% scholarship in tuition fees.

BITS, Pilani does not offer MCA.

Yes, you can. NEET 2018 Exam will be on 06th May 2018. So, not much time left for the preparation. For NEET preparation, you should now only study the most important topics. Solve previous year question papers and take online mock tests. Download previous year NEET Exam Question Papers here:
https://scoop.eduncle.com/neet-previous-year-question-paper
Focus more on NEET Physics section. This is the most difficult section in NEET Exam. Learn all the formulas as they are the backbone of NEET Physics. Make notes of all the important stuff. This will help you in your last minute preparation.

For best labs and faculty members IIT, Delhi is best. Then comes BITS, Pilani and in last DTU.

You have decent chances of getting a call.

Yes, you can do projects here. There are many clubs and technical societies that do such kind of projects: ANANT working on satellite, ACYUT and Robocon working on robotics, BAJA designing and making sports car.
Startup culture also supports lot of initiatives. There is entrepreneurship cell here.
Such kind of initiatives is engaged here.

Check this link to know about the courses offered:
http://www.bitsadmission.com/hdmain.aspx?id=1262015.

GATE STIPEND:
This stipend is given to after you have passed out of the course.
Under teaching assistant ship:
BITS stipend or fee waiver or both as BITS stipend & fee waiver.
Under both conditions, you have to give extra hours in teaching assistantship and you are not eligible for GATE stipend.
So, answer to your question is yes, in case you do not opt for both the options in teaching assistantship.

http://www.bitsadmission.com/hdmain.aspx
check this website to know about the courses offered.