Hey,

Yes you can register for State Quota as well as All India Quota counselling.

If you get a seat in the second round of All India Counselling and you join it then you cannot leave the seat and go for state quota seat. You can leave the second round seat without joining it, but there will be forfeiture of security fees as second round is not free exit. To know more about NEET counselling, check here https://medicine.careers360.com/articles/neet-counselling

To get a list of colleges in which you can get a seat, you can use NEET College Predictor https://medicine.careers360.com/neet-college-predictor?icn=QnA&ici=qna_answer