Yes, goverment universities take admision on JEE main rank.it may be score depends on university.

That depends on government to government, because for an example in Gujarat one year ago it was compulsory to give JEE exam to get admission in any govt engineering institute and now it is not only your 12th score and GUJCET score are prerequisite to getting admission.

The minimum eligibility criteria for government university is that you should at least have 75% in any board.

There are two type of eligibility criteria for taking admission in IIT and NIT. You will have to be in either any one of them. One is that you are in top 20 percentile of your board and second is you have scored above 75% from any board. If you are not lying in any one of them you can't take admission in IIT and NIT. But still you can appear in exam, and if you score good you will have many good college options.

All the very best...

You have to score more than 70 % to get NIT, IITs.