How do I make it clear I only accept cat friendly dogs?

I'd go with the VERY direct, polite. "Dogs must have a proven track record successfully living with cats."

You could not take dogs into your home, and only do overnight stays AT someone's home, thus no worries for Your cats. This of course depends on how comfortable you are with leaving your cats at your residence without you there.

You could also Do the multiple Visits over a day leaving the dog staying in their home. Obviously you would need to explain you wouldn't be there overnight.

It may help to use that as your tag line although it will affect the amount of customers who choose to book with you or go to your profile. The other option is to politely mention it from the get go after you receive the initial request.

Yes, your search ranking will go down if you continue to decline bookings, and the reason you may be getting so many requests is because your rate is so low in comparison to others in your area.

Unfortunately, there is nothing you can do. Customers can't be counted on to read your profile for important information, even though you have highlighted it at the very start. Also, they're generally unaware how their dog might react to a cat if he/she has never been in contact with one. Cat-friendliness isn't a filtering category, like the dog's weight.

Try some stronger, more descriptive language since people may assume their dogs would be okay with cats. For example: Does your dog really like cats? If so, then this is the place! If you don't know, then it probably isn't."

Good luck!