JEE Main

It is very difficult to get NIT-trichy. You can try applying for colleges in rourkela, durgapur etc with easier cutoffs.

Yes, you can, but you have to get into premier colleges like good NITs, DTU, NSIT and IIIT, otherwise it may create problems if you join a local college.

It will be hard to get any state government college as even the counselling of many colleges is about to end and the sessions will probably start from mid of August. But, you can still look for it in your area. But, the best option will be to join any private college because your board percentage and Mains score is too low. And, the private colleges are doing exceptionally well, so you can go for it.

For OBC category, 180+ would be safe for CS branch. Also, it depends on your state, whether state quota or all India quota. But, top NITs are NIT Warangal, Surathkal, Srinaga and Durgapur.
All the best.

There will be no effect on your placement but at least you will do much hard work to get next year and achieve better rank than this year.

First, complete the NCERT book then study the coaching material. Test series comes in last.

If your general rank is 6213, then u'll easily get IT at IIIT Allahabad because of reservation quota.

Nowadays coaching is required for the preparation of all exams. But if you do not want to join a coaching then you should focus on what is taught in school and attend the classes regularly.
The school will prepare you for the boards. for JEE Mains you should thoroughly prepare your school syllabus of class 11&12 as the syllabus for JEE is same. You should solve previous year question papers of JEE and can also buy some books for JEE preparation. Your basics of 11th and 12th class should be clear.
All the best.