The house is in a terrible state of disrepair and I am not sure what else will come up once walls come down and floors lifted - as it is many of the walls show signs of moisture and I am worried there may be some mold. 

Not a lot of course but puddles here or there due to moisture in the ground.

We really need to get that moisture out of the bathroom. 

Some foundation companies can provide this in a less official manor but as long as you have the map then you can see what needs to be done (example picture attached) Water Flow - usually the foundation has shifted because moisture is not being properly regulated.  

To continuously have your property moisture stable, is a good thing.Your A/C will last longer with less use, but this does not mean it needs to rest.As stated already, if your unit never shuts off, the issue is less with your tenant and more with settings or the envelope of your house.

If a tenant brought this up with you, it could be a good sign.Cultivation is a bigger issue because of the moisture and lights use since it has to be inside here. 

When painting brick you really want to make certain you a) prep and clean the surface so the paint adheres properly and b) make certain you use a breathable paint that wont trap the moisture inside the brick/building envelope behind the paint.

Moisture is the enemy of plaster.

Foundations should never be backfilled with clay due to the moisture retention. 

The concentration of the moisture was low- much less towards ceiling and wetter towards the floor.