PA28 Flight Instructor with student reported the rudder pedals malfunctioned after takeoff and did not respond properly in flight. Alternator also failed and Instructor returned to departure airport and landed safely.

Narrative

Aircraft was located at ZZZ after alternator failure and repair. I was asked to go pick it up with my student. During preflight I noticed the rudder was moving just a little more than normal. I called Maintenance and described the issue; we did a simple check with the peddles and made sure the rudder was moving properly with pressure on each pedal. I have flown aircraft with a rudder that moved slightly and Maintenance also assured me that some movement was normal. After getting an OK we filed for IFR; the clouds were at 2900'; we conducted normal startup and runup procedures. During taxi my student said he was using more pressure than normal on the left side. I took controls to check the pressure and again; it felt normal with just a slight pressure. I gave controls back and at each turn inspected the rudder itself while my student conducted the turns and I monitored. The rudder was moving freely and properly with each input. Because of my personal experience with aircraft needing small amounts of pressure on pedals; the phone call to Maintenance; and my own observations I determined that the flight would be safe. After receiving clearance for takeoff we taxied onto the runway and the student conducted a normal takeoff with me shadowing him. As our gear left the ground we gradually started losing rudder control and the left rudder pedal extended fully towards the pilots seats. My right foot was at the normal position with my left foot 6 inches above it. I took controls immediately and called Tower asking for a lap in the pattern and landing on Runway XXL with loss of rudder control. Tower cleared me for left traffic and asked us our situation. I replied that we had lost rudder control and would like to land and taxi back to Company Ramp. At this time I had not [requested priority handling]. I began to attempt rudder inputs; right rudder seemed to be working and when depressing left rudder fully to the forward position it also began to respond. The pedal however did not respond for the full 6 inches of motion before reaching full forward. It was at this time just 20-30 seconds into the flight that I saw our annunciator panel lighting up with low bus voltage and alternator inoperative. At this time approximately 30 seconds into the flight I called Tower and told them about the additional failing equipment and [requested priority handling]. One lap in the pattern for XXL left traffic and we landed safely and softly. The pedal returned to the normal position beside the right pedal. After a taxi to Company Ramp; we shut down and gave the plane back to Maintenance. I talked briefly with rescue services; giving my four required documents and a summary of the events. After the fact; it was determined that Maintenance located at ZZZ had broken one of our steering rods by oversteering our nose wheel during towing operations. The alternator and battery were still not fixed.