Flight Instructor reported a NMAC in the traffic pattern and taking evasive action to avoid another aircraft who entered in the opposite direction pattern.

Narrative

My student and I were coming into land Runway XX at ZZZ in Aircraft X; approaching from the North at 2;000 feet. I was making radio calls every 2 miles informing the CTAF of my position and intentions. I was using my portable ADS-B as well as visual scanning for traffic avoidance and saw that Aircraft Y was entering the traffic pattern on left downwind for XX from the south at 1;400 ft. MSL (Traffic Pattern Altitude); they were making appropriate position reports on CTAF. At this time I was at 2;000 feet MSL over the field maneuvering for the teardrop procedure to enter the left downwind for XX after Aircraft Y. There was another plane following Aircraft Y called Aircraft Z from the south at 1;700 ft. MSL as well who was not talking on the CTAF. I instructed my student to make a climbing right turn to 2;500 ft. MSL to avoid a head on collision with Aircraft Z; as we were within 300 ft. vertically due to them being too high while entering the traffic pattern (if we continued at that heading and altitude). After Aircraft Y entered the left downwind for XX; Aircraft Z entered the right downwind for [Runway] XY. I informed Aircraft Y that there was opposite traffic in the pattern. Once Aircraft Z entered final for XY they came on the radio frequency apologizing that they hadn't been on the correct frequency. I informed them they were operating on the opposite pattern and that left pattern turns are standard at ZZZ. At this time Aircraft Y was on final for XX. At this point I (Aircraft X) am 2 miles south beginning to turn back to the airport for the left downwind for XX. Another plane (I don't remember their tail number) followed me into the left downwind for XX. Once Aircraft Z realized they were on the opposite pattern they flew at 300 ft. AGL while sidestepping Runway XY to the left and climbing; they informed us they were going to fly and extend right downwind for XX to allow me (Aircraft X) to land. At that point I was established on left downwind for XX. My student continued their visual approach to land; we made a go around decision on final because our approach became unstabilized due to time and focus diverted to avoid Aircraft Z maneuvering in a non-standard fashion. I believe this issue was caused by Aircraft Z nonstandard entry to ZZZ pattern in combination with lack of radio communications; and loss of situational awareness for other aircraft in the terminal area; creating an unsafe environment for all aircraft involved.