Flight Instructor reported their solo student experienced an aircraft door opening after takeoff resulting in aircraft controllability issues. The student returned to the airport and departed the runway upon landing.

Narrative

My student pilot was signed off for his initial solo flight at approximately 53 hours of training time. The student and I did a pre-solo check flight which went well. I completed the paperwork and hopped out of the plane. Student asked me to help him reach the gull-wing door; so I pulled it down for him. He grabbed it from me and I walked to a safe location to observe his solo flight. I had my handheld radio and a cup of coffee; watched my student start up and taxi to the run up pad for Runway XX. Aircraft Y announced they were northwest of the field; inbound to land XY. My student said he was taking off XX; so Aircraft Y announced their intention to fly a normal pattern for XX. After a brief pause (which later turned out to student neglecting to perform a run up check); he taxied onto the runway for takeoff. He applied full power and at rotation; the aircraft door on the right side came open. This caused the plane to turn left severely. I watched my student barely clear the trees to the left of the runway; at approximately mid field. The student radioed that he was having trouble; so I reminded him via radio to fly the airplane. The Aircraft Y passed overhead and I radioed them to say my student solo was having a problem (my student was no longer sharing helpful deconfliction data by radio since he was covered up with his problem). Aircraft Y said they had the plane in sight. After guiding my student back to the field (he departed west; felt lost; then had some engine trouble caused by a drop in fuel pressure due to the left turning tendencies caused by the open door); he crossed over midfield at approximately 300' AGL. Aircraft Y was now turning base for a normal pattern. I again checked if they had my student in sight. They confirmed that they did. My student turned a low; close right downwind for XX and I observed fuel streaming from the right fuel tank vent (again likely caused by the left turning tendency and the right wing being low). My student turned a right base and flew under Aircraft Y (which was now on a short final). My student in distress had to make a left 270 to re-intercept final and land behind Aircraft Y. My student landed on Runway XX and then lost directional control; departed the left side of the runway; and ended up in the muddy drainage area adjacent to the runway. He was uninjured; but the aircraft sustained damage to the nose-wheel and propeller. The aircraft was later recovered from the mud. Contributing factors:-my student elected to skip the run up and associated checklist; which may have caught the door problem.-my student did not apply sufficient rudder control to counteract the force of the door; causing the initial sharp left turn and associated engine issues.- my student became disoriented; compounded by the aerodynamic buffeting and engine roughness while dealing with a emergency situation-Aircraft Y did not yield to an aircraft in distress. As an instructor; I would have expected better behavior from an older-sounding (based on the radio voice) pilot flying a complex high performance aircraft. - the design of Aircraft X's gull-wing doors presents a loss of rudder authority in the open condition. I gathered that my student stayed at full power the whole time; which accounts for the left turning tendency (p-factor); but I think the door was impacting airflow over the rudder. While my student could have done the checklist and caught the door; other failures could lead the door to come open anyways. Perhaps the aircraft company needs to design a failure point (automatic/mechanical) or a manual cutaway method that would separate the door in the event of an inadvertent inflight opening.-the presence of an additional aircraft added stress to my student's workload and directly caused my student to spend additional time flying with an emergency condition.