Where does the equation of constraint below come from? I've tried to rationalize it, but the angle will be 0 more than one time as the string unrolls, even though y will keep going down (right?), not coming back to 0 (y=0 is considered to be at the point of suspension). I'd appreciate any help, thanks!
The angle is $0$ only when $y=0$ at the point of suspension. It keeps on increasing thereafter, although the multiples of $2\pi$ correspond to the same point on the disc.
The equation $y = a\phi$ is just identifying the arc length as the vertical distance $y$.