If the mirror can be moved horizontally to the left or right, what is the greatest possible distance d from the mirror to the point where the reflected rays meet?

the distance d will be greatest when the rays are incident. the reflected area and the diameter of the mirror from an isolated triangle. the distance d bisects the triangle forming two right angle triangle. where d is 1 leg, d/2 is another leg, the reflected ray from the mirror to the point of intersection is the hypotenuse

tan Q = Y / X

So, the distance d from the mirror to the point that the rays intersect is always equal to the distance between the mirror and the point that the incident rays should intersect.