Engineering Entrance Exam

Admission is based on your JEE AIR. Forms will be out soon. Keep a check on BIT Mesra -https://www.bitmesra.ac.in
All the information regarding admission will be updated on the given website. There you may find a separate application form for the admission to BIT,Mesra extension centers i.e. Jaipur, Patna and Deoghar. There you may fill your choices with the application fee.
Follow the link for more details
https://bitmesra.ac.in/Show_Admission_Page?cid=1&deptid=11&pid=27

IIT, Kharagpur is one of the top 6 old IITs and is very much in demand. Also, CSE is the most popular course in this time period. Hence, you need to have a rank within 500 in JEE Advanced (CRL) for getting a seat in CSE at IIT KGP.

To get into IIT Kharagpur, you need to score a rank of at least 9-10k. However, if you get ranks below 2-3k, your branch won't be under your control, because there are different closing ranks for different branches.

Yes, you have a chance to get admission in NIT Durgapur.
Select the branch in which you love to work and you have very much interest in it with curiosity. Don't follow craze and popularity. Trust me, by choosing your interested field would be best for you than any other fields.

If you have take entrance test conducted by the private institutes like VITEEE for VIT, BITSAT for BITs then you should go there or its better to drop a year.

With this rank, you can only join private colleges. You have an opportunity to get govt. colleges but not this branch. So, it's better to drop or go for private institutes like LPU, Thapar, LNMIIT, etc.

For admission in Maharashtra, you should clear MHCET which can give you an opportunity to pursue your studies from a good college.

Yes, sure if you have not cleared JEE Mains, it's better to drop 1-year or you can join a good private college like LPU, VIT, SRM, BITS, etc. If you have given the entrance exam for the respective college.
All the best.

Yes, there are chances of increasing the cutt-offs this year. Around 200 marks should be required in JEE Advanced for getting in CSE in IITs.