Got a boxer/pit bull mix and grew up with rotties.
If you aren't prepared to instill the fear of God in them, ur asking for a situation. Dogs are pack animals. They have to always know their role in the pecking order. The fact that the dog was walking ahead of her shows he wasn't trained. Ur dog is supposed to walk beside you. Right side. Short leash them and shock collar them until they do. And you determine the pace and direction of the walk. Their fear of you should be great enough to override their attention span and prey instincts. That means rigorous hours of training outside and positive and intense negative reinforcement every time that dog gets the urge to chase a squirrel or gets distracted by a smelly patch of grass or anything else.
And even with that training, you are STILL taking a chance particularly with certain dominant breeds.
Edit: Cats just DGFA. Just feed em and let em do whatever. They just think ur an oversized slow ass cat there to entertain them and if they associate you with food, they'll drop off some contributions to the cat clan when they go hunting every now and then.