I got failed in class 12th exams in 2016. Am I eligible to give the exam for IIT JEE 2017?

No, a gap year after the 12th will not negatively affect your admission at private or affiliated colleges in Midnapore. 
Most undergraduate colleges in Midnapore operate under the rules of Vidyasagar University, which explicitly allows students to take admission into 1st-year undergraduate cours

The B.Sc. Environment course is generally less competitive than courses like Computer Science or Biotechnology, so the chances may improve. However, admission cannot be guaranteed because merit lists change every year based on student applications and available seats.

With 69.33% in 12th Science, get

Yes, even a decrease of one mark will be implemented. CBSE 12th re-evaluation can increase or decrease in marks.

CBSE Class 12 re-evaluation is conducted in following way.

• Candidates must refer to the marking scheme and question paper.
• Applications must specify the questions challenged with reasoning
• Results of re-evaluation (increase or decrease in marks) are final and binding.

Only candidates who have applied for scanned copies of their answer books can
apply for CBSE re-evaluation.