Engineering

First of all, congratulations dear!
According to your good score in JEE, if you are from Sc or ST then you will get any NIT and if you are from OBC or general there few chances to get the NIT or IIIT. Then you can try for good government. College which you can get through the state counselling proces

Yes, you can just take entrance for the college you are interested in. If you clear, you're in.
All the best.

You might get some government college. Check Shiksha college predictor to get more information. All the best.

From my point of view, you should go for VIT as it is one of the best and reputed college and the placements are pretty good.

Coming to the placements at VIT Vellore, it is awesome, and even we got a place in Limca book of records in the latest released records. We get more dream offers that is above six lacs per annum. Mainly for this branch, we'll have an additional benefit of the post graduation degree in the hands, so

See, if you are a Delhi resident, then you can easily get the college at this rank but if you are of outside Delhi, then this will create problem for you.

Please visit SRM website to get exact fee of this year.
http://www.srmuniv.ac.in/admission-india/fees/tuition-fee/fees/tuition-fee.
All the best.

You can easily get any of the colleges of your choice with this rank without having any problem. Plus, you are a resident of Delhi.
This will act as a booster for you to get a good college.