C-172 Flight Instructor on training flight with student reported a runway incursion and a near miss with another aircraft at a non-towered airport.

Narrative

Holding short of runway XX at [taxiway] 1. My plane was running late to come back to our home airport where another instructor was waiting for it. The pattern was very busy; so in my mind I realized that if we were going to take off any time soon I would have to line up and wait right after touch and go traffic passed in front of us. I took controls from my student to make sure we moved quickly. After what I thought was touch and go traffic passed in front of our plane; I started taxiing the plane forward to line up for takeoff. I was mindful of the fact that there was another plane less than 2 minutes behind us coming in on final. I made a radio call and started moving onto the runway. As I taxied forward I realized that the plane in front was not actually touch and go but was full stop. They were slowly moving to the exit taxiway preventing me from taking off. At this point though I was already on the runway and could not move back to the taxiway. Traffic was still coming in behind me so with short time to think I made the decision to takeoff as soon as the traffic in front turned. During the takeoff roll the traffic on final called a warning me of their position and asked me not to takeoff. Then they said they are going around and side stepping to the right. By the time we heard these calls I was close to rotation speed; when I heard they were going around and sidestepping I decided that the safest thing to do in this situation would be to continue takeoff and remain low to avoid them. We communicated to avoid each other and were able to depart safely afterwards. My decision making was affected by multiple things. I was rushed to depart; and I had expectation bias when I thought that the plane in front of me was touch and go. I should have not let these things affect me. If I had come back late it would have been fine; and I could have used the time to ensure a safe takeoff.