Here I go, I will probably not be too popular but .....
Black snakes are relatively harmessless to humans and a 13 lbs. furbutt. They are very beneficial in keeping the rodent population in check. A black snake bite could cause a local infection, but other than that I wouldn't worry. It would take a VERY large snake to eat your baby and snakes don't kill except for food.
I would approach the neighbor with caution, you certainly dont' want to create an environment where you have an angry neighbor - who knows they might take their frustrations out on your furbutt.
You can treat your yard for fleas and ticks and use a snake repellent around the fence line.
I agree with Villette and Judy - an unsupervised Doggy Door for a small dog is a recipe for disaster. LOL just think if Callie has brought the snake in the house - I bet you and grandma would not have been too happy about that! I
Good luck to you