Indian Institute of Technology, Delhi

No you need to qualify JEE advanced to get admission in IITD. You need to get a rank in top 3000 in JEE-Advanced.

IIT Delhi is one of the top ranked institutes of India. They have very good professors and good research options. IIT Delhi is well known by US universities, so it will definitely be a plus for you when you apply for MS in MIT. Just make sure you maintain good cgpa and undergo quality research in undergraduate.

Each hostel has a gym along with a gym at Student Activity centre.There are facilities for lot of sports.

There are no issues of security. Each girl is safe there. There are lots of co-curricular activities like music, dance, drama and so on.

Try for core branches at old iits
Iit kanpur kharagpur bombay roorkee madras are best

IITs are good for all branches.

Its little difficult . But do not lose hope. You can take admission in other private colleges.

Both options are equally good & have good job opportunities. Production & Industrial engineers compete with mechanical engineers for jobs, because of this reason, I will like to recommend you to consider chemical engineering as your career option.