What does it mean when a ray intersects an object? Does the ray change when it passes through an object and if so, does it depend on the material of the object?

I think yes. There could be reflextion or refraction depending on material of the object. And that's where we use recursive ray tracing (trace from each point where direction changes and all the way to light source or some maximum depth). That's reasonable for me, not sure how they do it in real world.

What about edge cases where the ray falls exactly on the edge of the box? Would we just have the freedom to choose whether or not it intersects with the box?

@LuxuFate The only case I can think of where it would matter is if you had two axis-aligned boxes perfectly adjacent to one another, and you try to shoot a ray perfectly between them. There is no gap between the boxes, so the ray should be blocked.

This can end up happening if you have a scene made of boxes with perfect round vertex coordinates, and your camera is also perfectly oriented relative to those boxes.

In practice this would a pretty rare situation though.