Hello Rajat.
basically there are 2 types of courses on the basis of fee structure one is called self financed and the other is govt sponsored. Self finance courses are expensive then govt sponsored for obvious reasons, government bare the cost of your education.
But it does not mean all gove clgs are state sponsored NLU is prestige college but the courses is termed as self finance as the student pay all the cost.
In most of the govt clg which provide regular courses(BA BSc ) are state sponsored. So the fee which is paid by student is almost negligible it is usually the exam fee. 

Coming to your second question almost every llb courses in India have entrance exam. NLU the most famous institute in India takes its students on the basis of CLAT exam.
Many other clg providing LLB degree conduct there admission based on CLAT score. I hope you get your answer

All the best for your future