JEE

Yes. You may wait until the last round.

You can appear for JEE advance for only 2 times.

You can see the answer key of many coaching institutions released this day evening.
Here is a link for an online answer key by resonance:
https://www.resonance.ac.in/answer-key-solutions/JEE-Main/2017/jee-main-2017-AnswerKey-Solutions.php

First go for NCERT books of physics, chemistry and mathematics. Then go for NCERT exemplar books. After you have completed the both, go for D.C. Panday for physics and Arihant Publication's 8000+ MCQ for both maths and chemistry.

For VJTI, you need to score around 150-160 marks in MHT-CET to get any good branch.

If you have confidence that you could prepare well and qualify for SAT and JEE as well with good scores, only then choose dropping a year as if you don't get desired results after an year you'll be frustrated. So think wisely before you make a decision. Or else, there are many good private colleges

The minimum score to get a seat in NITs is 130-150.

Through this marks, it is difficult to get good NITs through this score. If you belong to general category, then it is not possible
So, please check in this link about the cutoff according to your caste category.
http://jointseatallocationauthority.in/josaa-cut-marks-iits-nits-scstobcgeneral/

These exam are not compulsory after class 12th. It depends upon the career option you want to do after class 12th.
If you wants to become engineer or doctor, then you may appear in exam, but if not, then it is not compulsory.