@Jennifer1897 I have both a dog and a cat. I make sure to mention them in the listing info, along with a caveat that anyone who is allergic to dogs or cats, or afraid of them, to please not book with me. I also plan to update my photo gallery with photos of them, even though no guest has ever had an issue.
I also mention the animals in my first message to a guest when they request to book (if you use IB, which I don't, you could message it right after their booking is confirmed, so if they have a problem with it, they are within the free cancellation window). Asking the guests if they read about the dog and cat also gives me an indication of whether the guest has thoroughly read the listing info.
I wouldn't make too much of a deal about reassuring guests that you keep the dog away from anywhere a guest goes, as you still don't want someone booking who's afraid of, or allergic to dogs. Because they'll still be afraid and allergic, even if the dog is behind the baby gate.
Many of my guests have assured me they love dogs beforehand, and have quite fallen in love with her and say they'll miss her when it's time for them to leave. Mine spends almost all of her time outside, even at night, and rarely comes in the house, although she is free to do so. Basically she barks at the guests when they first arrive, then once she realizes they belong here and they've given her a pat or two, she whines and acts really sucky when they come home after that, which seems to charm them.