Does the direction of rotation of the propeller affect the direction of the thrust generated?
It depends.

For that simple kind of a fan with flat blades, yes. For anything that was ever installed on a full scale airplane, it will generate no or very, very little thrust, more likely backward.

The blades of a propeller are airfoils—like wings—mounted with suitable angle of incidence and usually slightly cambered. Any efficient airfoil is necessarily designed for one direction. If you move them backward—by turning the propeller the other way than it's designed for—they will generate very little lift. And the remaining bit may be cancelled by the camber, because the incidence will be for generating thrust backward, but the camber will still favor forward thrust.
For much more details, see
Is there an aerofoil that gives reasonably good lift for both air flow directions: forward and backward?

Only flat plates do generate lift both ways, so a simple fan with flat blades would indeed generate backward thrust. But it's not efficient in either direction in the first place, so it may be appropriate as a ceiling fan, but definitely not as aircraft propeller.