Flight Instructor with student reported observing a NMAC between two other aircraft in the traffic pattern at a non-towered airport. The Instructor stated the aircraft that caused the NMAC later overflew the instructor's aircraft on final approach and then subsequently cut off the aircraft that was involved in the first NMAC.

Narrative

An aircraft entered the AUN pattern from ZZZ on a crosswind entry. There was an aircraft on the departure leg that Aircraft Y flew in behind and turned downwind. My Student and I were on a left 45; which we announced prior; and slowed down in order to provide spacing for the abrupt crosswind entry. The departure leg aircraft announced earlier they were staying in the pattern but instead had to depart and re-enter on the 45 entry behind my Student and I. We were now number 3 to land. Aircraft Y was in front of us and there was another Aircraft Z in front of him. The other Aircraft Z; number 1; turned base and then final. Aircraft Y then turned a tight left base and cut off Aircraft Z on final and entered a collision course. I radioed and let Aircraft Y know they cut someone off on final. They aborted their landing and continued straight ahead while Aircraft Z initiated a go around to avoid collision. Aircraft Y thanked me for the call and turned around; now on a right base headed directly for me; as I was now on left base and I had announced it before. I radioed him again letting him know my position and asked his intentions. He did not reply. We turned final and announced a touch and go. Aircraft Y had followed us on final and flown over us; entering a go around on top of us as we were now on the departure leg to stay in the pattern. I told him I didn't have him in sight and he said he had me in sight and turned a mid field left downwind. At this time he once again cut off the other Aircraft Z who had re-entered a left downwind to try and land after their go around.This was a good example of why AC 90-66B should be enforced in the training curriculum at non towered airports. Non standard pattern entry can lead to dangerous situations such as this. The Aircraft Y pilot consistently enters a busy traffic pattern on a crosswind leg; which interferes with anyone on departure; final or on a 45 entry. I debriefed with my Student the mistakes that were made and the key points of AC 90-66B and how a predictable entry helps keep everyone safe.