Flight Instructor on training flight with student reported a NMAC with another aircraft in the airport traffic pattern.

Narrative

Flying on the downwind on Runway XX at ZZZ after taking off already for traffic pattern work for training and was on midfield downwind with my student at 1500 ft. MSL. Making proper radio calls for traffic avoidance the whole time while also maintaining visual scans for any other aircraft. I hear a call on the frequency while on midfield downwind 'ZZZ traffic; Aircraft Y entering midfield downwind for Runway XX'... as soon as I hear the radio call; I start to slightly panic as to figuring out where this airplane is located at that was coming out of nowhere making that one radio call. I look out to my right side right below me; and this plane is passing me up on the right side directly below me. (Would have been a catastrophic mid air collision had I been at 1300 ft.) He passes me up and continues flying straight while also flying past the field. This plane was at 1300 ft. MSL. I then make another radio call 'ZZZ traffic; Aircraft X on downwind for Runway XX fullstop'. The aircraft makes no other calls I continue flying the traffic pattern and was scanning every part of the traffic pattern for Aircraft Y while also making proper radio call outs; no sight. I land safely and clear the runway. My student and I decided to do one more lap for practice in the pattern since we don't see this hazardous aircraft anymore. We take off and had Aircraft Z following us safely in sync maintaining separation from each other the whole time. I turn crosswind and make my radio call; Aircraft Z decides to take off and follow (we're still looking around for Aircraft Y by the way). I call for downwind turn as turning downwind; Aircraft Z calls turning crosswind. 2 seconds later Aircraft Y calls turning crosswind right after Aircraft Z. I make another radio call stating that I'm established on the downwind for Runway XX full stop. I start my descent abeam my touch down point and when I start turning base at the 45 degree point from the runway. I call turning base on XX fullstop. Aircraft Z calls 'on downwind for 16 fullstop' and right after; Aircraft Y calls 'on downwind for Runway XX' Before I turn final I ask 'Aircraft Y; state your position?'. 'Aircraft Y on downwind for Runway XX'. I click back and state 'Aircraft X turning final for Runway XX number 1 fullstop'. 'Aircraft Z states on downwind turning base for Runway XX number 2 fullstop'. Right after; I hear 'Aircraft Y turning base for Runway XX'. with no sequential number followed by it. I land safely on the runway and cleared the runway and decided it was a good time to call it quits and get away from the aircraft that doesn't know how to maintain separation properly. 'Aircraft Z on final for Runway XX'. Right after; 'Aircraft Y on final; doing a go around to give you some room to clear the runway'.This occurrence was a lesson for my student and I to always be vigilant of uncontrolled airspaces and you can never be too vigilant. I wanted to make this report because I don't want this pilot flying without proper training or traffic avoidance. How to legally enter the traffic pattern at an uncontrolled airspace. Also would like to have that hazardous pilot receive proper medical attention with hearing and vision check from an Aviation Medical Examiner to constitute if he is safe to fly or not. I am blessed to make this report since if we were at any lower altitude I wouldn't be able to walk out of the airplane today and raise awareness to what could go wrong in an uncontrolled airspace. While all this was happening; the owner of the airplane I was flying decided to track the airplane on the traffic pattern and screenshot the Near Mid-Air collision between me and Aircraft Y with our altitudes and flight path. I would like to have it sent to whoever will be able to receive it. This airplane had 2 instances where the pilot had posed a hazard; possible mid-air collision to result in a catastrophic occurrence. And also runway incursion with Aircraft Z. All 3 of these airplanes that were in this situation didn't have an accident due to me and the other pilot in Aircraft Z maintaining positive communication on the radio.