I have worked with companies that specifically are certified to remove mold and I was told not to disturb it or mold spores could get on your clothes and travel throughout the house.

You are getting disturbed at inconvenient times. 

Another common one was noise disturbances.

If I tell tenants there is a $50 unlock fee, they are more likely to hide a spare key or give one to a family member to call them in case of emergency.I agree that the examples given in that "landlord hate forum" are next level disturbing.

You can place "reasonable" restrictions on them, such as requiring a Veterinarian's health report; requiring spay/neuter to prevent aggressive behavior or commercial use; animal must be in a carrier or held when in confined spaces such as elevators and stairways, and leashed in common areas (unless this interferes with the task the animal performs, such as a seeing eye dog); handler must clean up after the animal in all common areas; handler must maintain sanitary conditions for the animal including no offensive odors emanating from the residence; animal must not disturb neighbors with nuisance barking or aggressive behavior; and you can require annual updates of both the "request for accommodation" letter and the Vet report.

You can perform a title search on any house, and no one will ever be disturbed by it.

The tenant across the street was mentally disturbed and depending on his mood he could be erratic and threatening.

I am afraid that this could be disturbing to the other two tenants in the building.

If one tenant is disturbing other tenants it is defiantly your responsibility to protecting your business by resolve the problems.

As long as it does not have too much traffic and does not disturb the other tenants you might not even know it is there.