NIT Trichy - National Institute of Technology

Since your JEE scores are very low and against the bihar board marks, it is almost impossible to secure admission without a reservation.

I dont think there is any chance of nit & iit based on jee advance so u r not elligible for iit

I am sorry but your rank is too low to get into NITT.

NIT-Trichy is one of the premiere institutes of India. If you ask specifically about chemical engineering, it is one of the best. About your second question, PSUs rarely come for campus placement. ONGC and BPCL have different off-campus recruitment policies and don't come to NITs or any other institute as well.There are some PSUs which come but the number is too less.

Depends on which institute you are looking to get admitted into. Old NITs and IITs have a decent cutoff. Exact figure varies form year to year and also your board percentile. Go through last year cutoffs to get an idea.

Its hard for you as you board score is less. Also depends on your current year percentile in boards. You might get in new NITs though. NIT-T, NIT-K etc will mostly not available to you. Also it'll make a huge difference if you are from a reserved category.

In India, the best is Institute of chemical technology,Mumbai. With you jee marks its really hard for you to get admission in NITs. If you are from a reserved category you might have a chance.

105 is the last year cutoff for JEE-mains.

The total expenses for tution fees per year will be Rs 114,000 per annum which includes all tution , library and etc . For hostel and mess includes Rs 53,000 per annum

There is no point giving a false hope. I would suggest start looking for admission in premier private universities like BITS, MU and Lovely Professional University etc.
Taking a drop would not be a logical step as gap year affects your career a lot.
Dont worry on the placement or Education part. If you would be good you would obviously do well. Best of Luck.