They are ceiling registers.
My guess on what is happening is the register is being cooled such that after a shower the humidity forms condensate on the metal register.
Additionally, if the register is closed, the air pressure results in leaks around the register periphery, which cools the textured ceiling surface (especially the points that project into the room) and, again condensate is forming.
There are two possibilities then. Water should NOT drip out of a register no matter what the humidity.
1). The AC unit is over sized and or the air flow over the coils is too low.
2). This is more likely. The furnace has not been cleaned and the condensate tray is full wet debris and not draining away to the house drain properly. These units need cleaning thoroughly every year.
In this part of the world where AC is used about half the year, and the heating gets used heavily the other half, pretty much everyone has their HVAC professionally cleaned and serviced yearly.
To service the burners etc, requres a licensed technician. So I can not touch mine. A home owner did in the Twin Cities a few years ago. He did it incorrectly and the furnace blew up and killed a neighbor. He got a long prison sentence for negligent homicide.
This is especially important here as we use propane, which is much more corrosive to the burners, than natural gas.
The service man is due here Monday, to clean the unit before winter.
If it is either of the causes I cited, and I strongly suspect one or the other, then after correction the ducts need professional cleaning.
The mold on the wall needs treating with bleach and then repainting. Painting is important and will prevent recurrence as long as the condensate coming out of the vent is stopped.