Flight Instructor with student reported an aircraft followed them closely in the traffic pattern and then landed behind them on the same runway before the Flight Instructor had exited.

Narrative

Entering ZZZ from the south; made radio calls that we were preparing to overfly the field and teardrop into the right pattern for runway XX. Aircraft Y made calls that he was back taxiing on runway XY - winds were reporting variable so we switched to XY. As we made our teardrop call; Aircraft Y was landing for a full stop taxi back. We turned final and Aircraft Y began back taxiing. Aircraft Y said over CTAF 'don't go around; I'll stay out of your way' but I did not trust the situation so I instructed my student to go around. Aircraft Y took off immediately after our go around and stayed extremely close to us the entire pattern - ADS-B said 0 ft vertical separation and directly beside us. I asked over CTAF if Aircraft Y had us in sight; he said yes but did not show any signs of trying to distance himself. I could visually see him just below us out my window and kept him in my sight. During this pattern lap; multiple planes reported tailwinds on final; Aircraft Y fought over CTAF that 'XY is the active' and would not let others switch to the wind favoring runway. On final; I was helping my student land with the tailwind and we rolled down the runway towards the XX end. Looking behind me; I realized Aircraft Y had landed on the runway with us maybe 1500 ft behind. Another pilot reported that Aircraft Y had said on frequency 'who else is on the runway? This is unsafe!' but had not made any attempt at a go around. After landing; another pilot made an attempt to discuss the flight with Aircraft Y and he shrugged off any claims that his decisions were unsafe.