A C172 pilot reported a NMAC in the pattern at a non-towered airport when another aircraft made a non standard pattern entry.

Narrative

I was approaching ZZZ on a left 45 for Runway XX; which was being used by multiple aircraft working the pattern. While I as at TPA approximately 3 miles from the field; I heard Aircraft Y say that they were approaching the field from the east with intentions to enter the pattern for Runway XX. I assumed that with an aircraft on the departure leg and myself on the left 45; that Aircraft Y would overfly the field above TPA and enter behind both of us. The aircraft on the departure leg extended to create safe spacing between themselves and my aircraft. As I turn midfield left downwind; Aircraft Y is bearing 290 and flies directly over the Runway XY numbers and turns left to enter the downwind. This cut off the aircraft that was previously on the departure leg and was now beginning its turn into the downwind. All of the radio calls from Aircraft Y were very difficult to understand and there was no evidence or claim of an emergency in the pilot's tone or words that could be understood. At this point; there are three aircraft on downwind with my aircraft being in front. I extend my downwind slightly to allow for a twin engine aircraft to depart. When it was safe to do so; I made my radio call and turned left base. Seconds after this; Aircraft Y calls that he is turning left base. He was turning a slow elliptical base turn that cut inside of my base to final turn. I told him on the radio that he needed to extend and turn right to avoid. I had to deviate to the right of the glide path and accelerate to create more spacing. With numerous radio calls from myself as well as other planes in the pattern telling Aircraft Y to go around; he still stayed on final 200 feet directly above me. I had Aircraft Y in sight and was ready and able to turn right to avoid his aircraft. He eventually did go around but stayed only a few hundred feet off the ground and flew over the east side of the field. He then immediately after turned a left base for Runway XY and made a full stop landing which forced another aircraft in the pattern to go around and avoid him.I just vacated the runway and was taxiing back when Aircraft Y landed on Runway XY. During the taxi back; the pilot of Aircraft Y claimed that he had alternator issues and said that he was in an emergency solely because of a possible alternator issue. Not only is an alternator issue in the pattern not something worthy enough to declare an emergency for; but nobody ever heard him say Mayday; Mayday; Mayday. There was no excuse for how he aviated; navigated; and communicated in a way that almost caused a severe incident if it weren't for other proficient pilots reacting to the situation smartly and safely.