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).