I don't want to charge my clients per dog if I'm just doing drop ins. How do I change my rate from being per day/PER DOG?

Change the rate for the additional dogs to $0

You can say what you charge in your profile, and adjust the price when you receive a request. (There is an "edit price" link with the request. When you click that, you see a "one-time price adjustment."

That's how I do multiple-day and dog discounts.