I'd probably rent to #4 and maybe 5-6 if they had a landlord reference that the animal isn't harmful to property. 

When tenants cross the line and use the "m" word, you send a notice that they move out effective immediately no harm no foul to protect themselves.  

Since you were able to replace them before you lost any rent, I'd put this one in the no harm no foul category and return their deposit. 

She is saying that she's had cancer (melanoma) and that the sunlight can be harmful.

My suggestion is to look carefully at anything that could cause harm in a furnished apartment.  

What harm are the shelves doing.

The realtor is responsible for letting people in an making sure they don't do harm while there.

If your tenant, or their guests (which may include children) or their dog, is harmed by a wild animal with sharp pointy teeth living under the deck with its growing babies, that they informed you of and asked you to get rid of - then I see "liability" on a neon sign hanging above your head.Use your common sense here, instead of looking for some technical loophole.  

In a nutshell, I came here with a technical question which involves balancing his moving out and my finding a new tenant, hoping to find the answer to guide me so I don't screw him, and not harm my business at the same time.

----Many times, when you think you're helping people, you're really enabling an unsustainable lifestyle, and enabling them usually does more harm than good over the long run.