Engineering Entrance Exam

No that can't be done as each candidate will be having a specific date and time which they would have already chosen during the slot booking. So I think, you cannot attend the examination as you have missed the date which you have selected.
Or you try to call the BITs admin regarding this query, but mostly you will not get a chance to take the exam again.
All the best.

You can admission in LPU, JSS Noida and CU. You can check for NITs.

First, tell me in which city and state you reside. There are some good colleges in Pune, Maharashtra. Also, research about the colleges in your City/Town or State. Dhole Patil College of Engineering, Pune you can get admission there. Go the website of the college i.e. www.dpespune.com Request a call back from it then fill the enquiry form on the website. Any Further queries, comment. Will glad to help you.

It depends on your rank. Also, DCRUST, Murthal accept JEE or not for this year. Did you appear for Haryana Engineering Entrance Exam this Year? It also depends on the rank the in State Engineering Exam. See the last year cut-offs of DCRUST, Murthal. Fill the JoSSA admission form properly. If DCRUST, Murthal is available in JoSSA's list of engineering colleges. Do your research.
Good luck.

Now, NITs considers JEE Advanced for the admissions. So, you should have a good score in JEE Advanced to get admission in NITs.