Getting asked about a companion dog for existing tenants

[Garen T.]

Received an e-mail this morning from my existing tenants. They have been there for 2 years and I allowed them 2 cats for which they had doctor notes for as companion animals. 

The letter today had the following elements. 
A tenant who has been there for over a year struggles with PTSD 
A companion dog would be a help. 
They would like a small breed dog like a Chihuahua 
They say that have training for dogs and can train the dog to not bark or be destructive
They say that he does have a medical allowance to have a companion dog

Questions I have. 
Am I legally obligated to let them have a dog?
Is a chihuahua really a companion dog? 
Can a dog be trained not to bark or be destructive (I've had dogs, I love dogs, I think dogs are great, but I think saying that they can train a dog not to bark shows to me that they don't have much training experience with dogs)
Can I let them do this, but charge a monthly fee and make it worth it? I know this is a valid strategy that is discussed here. 

This is a duplex and if the dog does bark, I'm sure I'll get complaints from the neighbors. 

Thanks!!