Flight Instructor reported a NMAC with another aircraft maneuvering around the pattern at a non-towered airport.

Narrative

I was flying to DMW from the west to enter the pattern when I noticed Aircraft Y also flying around the area. I tried to make several attempts to make contact with him but I could not hear him on my radio while I heard others on the radio. I was at 3000 Ft when I noticed they were significantly lower than my aircraft around 1200 ft. He was doing 360s around the area when no one else was in the pattern and I wanted to know his intentions. I had to fly further out to avoid him and I finally made contact with him on the radio. He seemed very upset and argumentative saying his intentions were to land at the airport. He continued to try to enter the pattern but then continued to do circles around the area at very low altitudes. DMW has a pattern altitude of 1600 and there was other traffic coming in. He finally did come in for a landing to do a simulated engine out and then did a very low go-around. He proceeded in the pattern a few more times and went to ZZZ. On the way to ZZZ he was directly behind me at my 6 o'clock and only 100 feet below me. I received a traffic alert from ATC and told them I was climbing to avoid the traffic and they also saw me. I had to climb 700 feet to feel safe around this aircraft.