Some want it run the first day of the month and others want to wait until their first payday of the month and we accommodate that as well. 

At first, I tried to accommodate them, but after a month of complaining, I gave them the option to move out with 30 days notice so I could find a new tenant. 

Exactly what she offered and you're accommodating a hard to place dog.

Free accommodation for the rest of the tenants life?

It's up to individuals to be reasonable, accommodating, and understanding.

I rent short term furnished accommodations within walking distance to downtown Seattle. 

If they ask for a reasonable accommodation based on a disability, they have to meet the definition of disabled and the request must be reasonable.  

In response to a much too lenient (in my opinion) proposal to accommodate her situation, she gets bent out of shape.  

@Paul Steward show me where ADA requires someone to undergo significant expense to accommodate?

If you intend to maintain a no-pet policy as an accommodation for someone with severe pet allergies, it would be smart to have them provide a note from their doctor verifying their need for a pet-free environment.