Flight Instructor reported that during landing rollout the student lost directional control of the aircraft through misapplication of the rudder pedals and locking up the brakes. The Flight Instructor regained control of the aircraft and safely exited the runway.

Narrative

My student and I were doing touch and goes at ZZZ; on departure on the crosswind to downwind leg at approximately 1;000 feet AGL; I reduced the power to idle and simulated an engine failure. I told my student to make a power off landing and make it a full stop. He glided the aircraft back to the runway and landed normally. Immediately after landing my student started to lose directional control of the aircraft and started to oscillate on the rudder pedals. I then instructed my student to regain control by applying equal and consistent pressure on the rudder pedals; I was also on the pedals assisting him. We started to approach the halfway point on the runway and I told my student he needed to start braking. He then pulled on the handbrake excessively hard which caused the wheels to lock up. While the wheels were locked up; he proceeded to start oscillating again with the rudder pedals. I took controls and audibly told him 'my controls'. When I pushed on the handbrake forward to release the brakes; he bumped into the throttle and advanced it; at this point we started to accelerate towards the end of the runway; I immediately pulled the throttle back; regained directional control and applied the brakes consistently/smoothly. I was able to safely exit the runway at the last taxiway. After this incident; I had a meeting with the owner of the flight school and we decided to drop this student from the program. This was not the first 'incident' with this student. He had flown previously with the assistant chief pilot and other senior instructors; after they hit a 'wall' with him in training and he was no longer progressing; I took over. Of the approximately 6 flights I had with this student; 3 had 'incidents' where the student had loss of control and if I did not take controls I believe it could have led to a fatal accident. [The student is in the] pre-solo phase of training with about 105 hours of dual received. The next day we did a debrief on the flight and when I brought up the events; he interrupted me and said he disagreed with it; when I asked him what he meant by it; he stated that if I wasn't there he would have recovered safely and that I was making him nervous.