my answer for Q1:
"If the ray hits a vertex, it dies". I take this property as the starting point of my solution
Let two neighboring disks D1 and D2 contain two such incongruent equireflective polygons. Let us take a vertex V1 of polygon P1; all rays hitting V1 will die; same behavior for the corresponding point V1' in disk D2, no matter if V1' is a vertex of P2 or not. All rays hitting V1' die, meaning that at some point they hit a vertex of P2, vertex which doesn't necessarily have be in V1' position. But is has to be somewhere on the ray support line, maybe before or after V1'; So this line contains for sure a certain vertex of P2.
This property holds for an arbitrary ray hitting V1'. We can have infinitely many rays hitting V1', each containing a vertex of P2. But P2 has a finite number of vertexes. So we have an infinity of lines - all convergent in V1' , each containing a point from a finite set of points. This leads to V1' itself being a vertex of P2.
Repeating the same procedure for each vertex of P1, we get that the corresponding point in D2 is also a vertex of P2. So the set of P1 's vertexes is included in the set of P2 's vertexes. Now we start from disk D2 with the same procedure, to obtain that the set of P2 's vertexes is included in the set of P1 's vertexes. The two sets are equal. The two polygons have their vertexes in the exact same (geometric) location on their disks - so they are congruent.
