Also, an Air conditioner will work more efficiently if there is less moisture in the air (Houston, Florida, Alabama, etc).  

They can also be a liability however if there is a moisture problem that cannot be easily solved.

Something about moisture transfer, but he showed me where bits of mold are starting to form on the roof beams.

I informed them that the reason mold keeps reappearing is because of the moisture in the room when showering.  

Tenant agrees to allow Landlord timely access to premises to do maintenance and repair.Tenant will notify Landlord in a timely manner of any damages or problems as they arise, including damages caused by Tenant, malfunction of appliances or fixtures, and water/moisture leaks and spots.Tenant will take care of the premises, keeping them clean and clear of debris.

Washington, lot's of moisture there? 

Buy a moisture meter and live by it.

At this point, the power has been off for over a year and no attempts to manage the moisture (even with a regular dehumidifier) have taken place in atleast 3 years.

I send handyman over to inspect the situation and he said the mold is literally only where the bed was but he (and tenant) think there might be mold behind the sheetrock and arrange for him to come back Monday (tomorrow) morning to remove the sheetrock, check for moisture behind and replace/repair.

Turned out the attic had no ventilation (due to the age of the house) so all the moisture from decades of rain and such covered the attic with mold.