Hello aspirant!
Hope you are doing well!

For receiving a call from counselling, a candidate need to have qualified NEET2020 with the minimum required percentile. So as to answer your question yes a person of NEET can participate in All India Counselling and state counselling(for 85% quota seat).
Mopup round is for those who could not get a seat in Round1 and Round2 of counselling Basically for counselling of seat allotment in medical colleges by states against the reverted seats from All India 15% qouta.

So it gives you a opportunity to obtain a seat in government college irrespective of lower marks than state cut off.

Hope it helps!

For more information you may visit https://medicine.careers360.com/articles/neet-counselling