On a PA 28, the fuel tanks each have a metal 'step' just inside the filler cap opening. The fuel level, relative to this step, is the determining factor as to how much fuel is in each tank. The fuel gauges on such aircraft are usually highly unreliable anyway and sometimes (?) are not even the correct ones for the aircraft. Lesson 2 would have been a GF excercise (?) so Mark 1 eyeball on the fuel level is really quite normal. Never assume that because the aircraft has only flown for, say, one hour since full tanks, that the fuel remaining is sufficient for, say three hours, always check the levels.
As to tyres on training aircraft, well that's a bit of a joke really. If you can't see canvas they're probably all right for lesson 2 when the instructor would be the one really landing the aircraft. If you were lanunching off into solo consolidation which presupposes some degree of multiple landings, then a bald patch might be rather more cause for concern.
I would have expected your instructor to have discussed both these points with you. I really do hope he did not just say: 'That's okay' without giving you his reasons.
