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.

enter image description here
(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).