NEET exam helps you to take admission through both All India Quota seats, Deemed university seats and also in State quota seats all at once.

Medical Counselling Committee (MCC) is the authority to conduct the NEET-UG counselling process for AIQ seats. https://medicine.careers360.com/articles/neet-counselling you can go through this link for detailed process of counselling.

Now, if you are opting for AIQ or DEEMED counselling you can easily also simultaneously opt for state quota counselling.

But, related to your second question, after any round when the next round choice filling options are given there are two options are whether you want to retain the alloted seat or want to upgrade. Now, if you want to upgrade then returning back to previous seat is not possible, only if your choice for upgradation has been locked; otherwise previous seats will be available only.