Well she left all the windows closed so the added moisture from the rain with no air flow caused mold/mildew to cover her stuff for the same reasons attics have soffits. 

It makes the two hours informative and fun.All of the documents and information booklets ("Mold, Moisture and Your Home", and "Protect Your Family From Lead In The Home", "Fire Safety", etc.) are given to the new tenant in a nice looking folder, along with our contact information and contact information for the utility companies.

Kyle:We make extensive use of closed cell foam as well - when renovating old balloon-framed houses it is the best/easiest way to air seal the walls and attic space.Applying spray foam directly to the underside of the roof deck turns the entire attic into a conditioned space, which means you need to close-in (if open) and insulate the soffits as well to have continuous insulation coverage up the walls and over the roof.However, you very well may still need a drainage/ventilation plane on the outside of the roof deck to ensure the deck material (board, plywood, OSPF) can dry out after absorbing moisture from rain/snow or condensation (by applying closed cell foam to the underside of the deck, you are changing the dew point in the roof deck construction).  

There are acoustic dampening subfloor material that will seal the moisture and dampen noise. 

Also thinking of buying moisture meter to check reading.

No, I won't put cameras in bathrooms :-) Don't want to go to jail :-) I can of course put moisture sensors there. 

After pressing her if there is any signs of moisture, water intrusion, or any visible mold ANYWHERE - I was told no, but that the tub had some black "stuff" around the perimeter (me - likely mildew...), but it has been there since she moved it.The tenant notified me that the basement in the building has " flooded" recently and she's concerned that there is mold in the air and got into her unit (4 stories above...).

The company that installed the roof back in 2015 came out and tested the walls & ceilings for moisture.

But if they turn off the light, it cuts power to the fan too, and the moisture maybe not be fully evacuated by then.So I educate the tenants- fans remove moisture, prevent mold.

You should buy a moisture tester and test the moisture of that beam.