Hello Aspirant ,

Yes you can attend both counselling processes . The dates of the two will never be the same , always the first round of All India Quota will take place first, followed by the state counselling first round. If you get your desired seat in  All India counselling first ,then take it ,but in case you do not like it , and then later after attending the state counselling if you get a better seat, then you may leave the AIQ seat and choose whatever you got through state counselling . The fee to register for government and private college counselling will vary from 10K to 50K. Also fill only those colleges you are interested in because when you withdraw , you security deposit money will be forfeited.

Use the college predictor tool to know all your options :

https://medicine.careers360.com/neet-college-predictor?icn=QnA&ici=qna_answer

All the best