I have a bathroom on the first floor of a two-floor house where the popcorn ceiling has dilated and cracked open twice within a year, and some mold is always coming back next to the ceiling, so I need to install a fan there. That bathroom has heating in the ceiling (therefore I'm avoiding ceiling fans), so I want to make a hole on the upper part of one of its walls -- one that's shared with a closet located inside the bathroom -- and install a fan there to direct the air downwards with a duct through the basement, where it would be exiting near the dryer vent with the necessary ending to avoid condensation and being a way in for animals. Does that seem like a good plan?
Most wall mount ventilation fans are intended to go thru an exterior wall to a cap.
Your plan will work, if you make allowances to support the duct away from the wall a bit, to make the initial turn down. Use smooth duct and metal tape for sealing duct sections. update Be sure to slope all horizontal runs of the duct such that they will drain any condensation to the outside.