Trainee pilot with instructor reported taking evasive action to avoid a NMAC at a non-towered airport.

Narrative

I was acting as PIC for Aircraft X; flying from ZZZ1 to ZZZ2 to ZZZ to ZZZ1. I was working on landings today in preparation of my Commercial Checkride; with a CFI. I was out from about XA:00 to XD:00 UTC; and this flight was conducted under VFR conditions in my local practice area Northwest of ZZZ1 Class C airspace. I arrived at ZZZ; around XC:00 UTC. I was working on landings. Me and my CFI were in the traffic patterns; safely flying and looking for traffic; both on our Garmin G1000 and outside the aircraft. We went on until approximately XD:00 UTC without incident. About XC:00 UTC; there were only about 3 planes in the pattern at ZZZ to my best memory; when Aircraft Y approached the field. They remained at pattern altitude;1800 MSL; while not joining the pattern. Though UNICOM was busy for this airport; I noticed he had made no position calls; or any radio calls for that matter. I let my CFI know; and we made sure to make visual contact with him; if possible. Eventually; we made a few more landings and were in the left downwind for Runway XX. Another red-colored aircraft was behind us; of which I did not get the tail number of; and was on the crosswind for Runway XX. Aircraft Y then made the hazardous move to continue directly in the ground track of the ascending plane; without making position calls. I then made a radio call to this individual; asking Hey Aircraft Y ; are you joining the pattern?" No response came. I made visual contact with both the Aircraft Y in question and the pilot of the red aircraft; who had to make the decision to make an evasive maneuver downwards to eliminate the risk of an impending collision. They were both at the same altitude on ADS-B on the G1000; and I saw them fly over one another; likely less than 150 feet clearance. Thankfully; no collision occurred. The pilot flying the red aircraft then made a call; frustratingly requesting "Aircraft Y you need to make sure to look next time"; before exiting the pattern. We continued for about 5 more minutes; with Aircraft Y in the traffic pattern. He made approximately 2 calls; both in downwind. After another touchdown at the 1000-foot marker; I exchange flight controls to my CFI; who takes off Runway XX once again; for him to do a landing. We noticed once we took off; Aircraft Y took off when we were approximately 1200 feet MSL; when we retract flaps. We turn to crosswind. Once we turned from crosswind to downwind; still not past the threshold of Runway XY; Aircraft Y turned directly into the downwind; completely skipping the crosswind and ascending directly towards our aircraft. No calls were made. I let my instructor know about this; saying "this guy is flying directly towards us." I make visual contact with the ascending Aircraft Y; who is now about 400 feet from us; now crossing our flight path. My CFI increases the power from 50% to 85%; increasing our indicated airspeed from 100 to 115 or so. We managed to land without incident; and rejoin the pattern. He eventually lands as well and taxis to either the FBO or parking. We make about 4 more landings before heading back to ZZZ1. We depart the pattern; and I make the decision to note the tail of this aircraft; and a brief summary of the incidents. As we do this; the CFI has control of the aircraft. I then noticed an unmarked aircraft on the G1000. I make visual contact once more with Aircraft Y. I verified it was Aircraft Y from earlier via the white color; make and model; and tail number. I also verified the flight path. We depart the pattern; avoiding him. We head towards ZZZ1; direct to the ZZZZZ fix and joining the ZZZZZ arrival. We notice that Aircraft Y is inbound to ZZZ1 as well; about 1500 feet below us; who are at 3500 feet MSL; about 5 NM away. We contact approach; and fly the arrival as published. During this time; we notice this aircraft going to ZZZ3. We observe on the G1000 one aircraft in the pattern; with their nose pointed to Aircraft Y . This aircraft was a Piper. They were making all necessary maneuvers to safely fly the traffic pattern. Aircraft Y was at pattern altitude; 1900 MSL; the same as the Piper. Aircraft Y flew toward this aircraft; to a point where they also had to make evasive maneuvers. Me and my CFI watched all this on our G1000's ADS- B tracker; and made the decision that a safety report would be necessary; as this aircraft seems to be a risk to people inside the same airspace as him. I am unsure if the pilot of Aircraft Y had any aeromedical problems; as when he did talk on the radio; the 2 times he did; he seemed to sound like an older man. Perhaps it is aeromedical; or not holding proficiency in flight maneuvers. I am sure he is a good man who may have had a bad day; but I am certain this report was necessary to ensure the safety of people in the air and on the ground. I wish not to withhold this information; and for something unimaginable to happen; when intervention can occur now."