Do reflected and incident rays interfere?

In a more realistic way you would draw wave beams instead of rays, like in the image below.

And here is the same situation as an animation.

_{(animated image from Wikimedia - File:Internal-reflection.gif)}

So you are right. The reflected wave and the incident wave indeed interfere with each other. Where the incident and reflected beams overlap, we get kind of a standing wave pattern. Where there is only one beam, we have just a simple wave.

Drawing rays instead of waves is just a convenient simplification. This is valid because waves propagate in a straight direction (like rays), and waves get reflected at surfaces (also like rays).