Flight instructor on training flight with student reported a NMAC with another aircraft in the airport traffic pattern.

Narrative

I was serving as an evaluator of a Certified Flight Instructor applicant. The applicant was returning to the airport of origin and properly established two-way radio communications with the control tower. Initially the applicant (who was the pilot in command) was told to expect a right mid-field downwind entry to the Runway XX traffic pattern. As we got close to the airport; the control tower instructed us to plan on a two-mile right base to Runway XX.As the observing party; I had to make all radio calls as the pilot in command's headset was not able to communicate and had to be plugged into a rear seat in order to have intercom capability between the two pilots on board.After acknowledging the most recent instructions; the tower then instructed us to enter on a left midfield downwind. As we were approaching from the East; I asked the Tower to confirm if he meant 'left' or 'right' downwind. The Tower then instructed us to enter a right downwind.At the same time the Tower was anticipating a handoff of a C441 from the ZZZ Tracon. At some point that aircraft was being vectored by someone; and the first contact I remember was the tower asking if the C441 was lining up for Runway XY or XX. A very short time thereafter (seconds) the Tower called us and informed us that traffic was 12 O'Clock and less than a mile. After searching diligently(as the Sun made viewing from the 12-5 O'clock directions difficult) I saw the C441 and immediately took evasive actions as did the C441 which appeared to be diving. My best estimate is that we passed within 500 feet of each other and quite likely might have collided.After this; we were instructed to follow a Baron who was already on final. Another aircraft was then instructed to follow us. During all of this I was having trouble with my headset jack and on short final asked the tower to confirm our landing clearance; I then promptly asked the pilot in command if the pilot in command had heard if we were cleared to land; and the answer I received was 'yes' within seconds the pilot in command landed the aircraft. The practice at this airport is to turn off Runway XX onto XZ where upon I asked if we were cleared on to XZ before turning on to it. We received our taxi clearance and heard no more from tower regarding our landing. I am still not sure if we ever had an actual landing clearance.It was fairly hectic at a tower with one controller; several aircraft trying to land; at least one aircraft making an apparent go around after possibly lining up on the wrong runway. In the confusion of it all; coupled with an aircraft whose radio reception was becoming difficult to understand; I am afraid that the pilot in command may have landed without a landing clearance.