General aviation Flight Instructor reported a near miss while in takeoff position on a runway with an aircraft turning final approach to land on the opposite runway at a non-towered airport. The pilot waited for the traffic to go around then safely performed their takeoff.

Narrative

The pattern was busy at 1R8 with Aircraft Y landing Runway 26. They extended their downwind so we could enter midfield behind them. Student pilot and I integrated into the downwind and followed the traffic - no issues. We landed full stop and taxied back full length for 26. Aircraft Z landed 26 and taxied behind us to refill and a new airplane (Aircraft A) entered the pattern. (This aircraft was having radio issues all day). Both aircraft were making a pattern for 26. A general call; 'inbound for Runway 8' could be heard on the radio. As this frequency is a shared frequency and two aircraft are making routine pattern 26 calls; student pilot and I did not believe the call was relevant to 1R8. This would mean the caller was completely unaware of the flow at the field. 1 of the aircraft landed 26 and taxied off for a taxi back. The other aircraft was midfield downwind so we made a call 'bay… traffic; tail number departing Runway 26; bay…traffic' and lined up on 26 for a short field takeoff. Brakes held. Aircraft Z appears from the tree tops of the south; at the midfield; hard banking for the last half of Runway 8…head on with us. They see us and say; 'looks like there is someone on the runway; going around' once clear of our horizon we released brakes; took off; and ran the pattern for another touch and go. While in the downwind the 'go around' Aircraft Z landed behind us. No issues.On the touch and go; we were in the departure leg about 800 feet when one took off behind us and we could see him below and to the left about 500 feet. No issues.We would also get in near proximity again during ground reference at a nearby field. We would maneuver south and west to avoid. No issues.