GATE

Are you qualified?
you may apply for DTU. Please check the website for more detail. Generally DTU asked for Only GATE so you may apply for it. Check regularly website gor dates etc

You may apply for it in different colleges. Many colleges do not ask for NET or GATE. They may have their own procedure of selection. Even you mag apply for several private colleges.

You should apply for NITs. They may consider this gate score. However you may also try for Jamia Millia Islamia, Delhi Technical University and Aligarh Muslim University, they will take a separate entrance test also.

Generally, a rank under 700 for general, under 2000 for OBC and any rank for SC/ST candidate is the key of getting admission in old IITs. However, slightly higher rank students can also get admission in new IITs.

GATE 2017 was easy as compared to 2016 and 2015. So, you can expect the cutoff doe Chemical engineering to be anywhere between 34-38 out of 100. Having said that the ranks would fall in this order approximately:
Rank Under 100 75+
101 to 250 65 to 75
251 to 500 55 to 65
Rank 500+ Less than 55

You need to spend around Rs 2.9 lakhs a years excluding hostels. It is worth doing because VIT provides you a great exposure.