Which nlu at best can I hope to get?

See it's always difficult to predict the college as every year the trend changes. It all depends upon the type paper(difficulty level), students performance and available seats. so if right i list some colleges also  that would seem very meaningless as u may or u may not get it.  Don't worry just register for counselling and let's see what best you get.But you'l get a NLU for sure

Do I need to belong sc/st/obc caste for RMLNLU to accept my domicile certificate. I'm a resident of UP. So I was just guessing if it could help in the admission process.

And u keep saying to register for the counseling. It's not 10 June yet. I'm supposed to register once the cut off list is out on the portal or am I wrong?

yes yes of course  when i say register for counselling i mean that whenever the registration opens.