Computer Science and Engineering

You can easily get admission in NITs like Agartala, Jalandhar, Shibpur and many more. You can even easily get admission in IIITs also which is a better option.

Sorry to say but at this rank you won't be able to get CSE in DTU. Seats in DTU is filled up at top 4000 rank.

Check the below link for colleges based on your rank:
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor
https://www.shiksha.com/b-tech/resources/jee-mains-college-predictor?utm_source=connect&utm_medium=CR_JEECP&utm_campaign=8030745
Also, do check for state engineering colleges accep

No, it's absolutely, not a good option. After doing engineering with CSE this is most important to get placed from the college campus. Otherwise its difficult to search job. Choose the college in which max companies are visiting the campus. So, finally, Vivekananda College is not good for CSE.

JEE Mains 2018 college predictor:
Use your JEE Mains 2018 rank to see where you can get admission at the link provided below:
See the courses, colleges & rounds you may crack
The result is based on admissions data of JEE Mains 2017
Get resources, insights on colleges & courses:
https://www.shiksha.com/b-

I would suggest doing B.Sc. As nowadays, IIMs do not prefer B.Tech or BE degree holders.
Even there might be a discrimination against you during the interview due to an engineering background, but it's okay if you want to make your career in the IT field, all you need to do is have some work experien

No, it is not necessary. You start a new career after 12th class in a new field.