Sorry, I am afraid I do not fully understand this question. Can you be more specific?
You wrote "My dog and I" - so "Моя собака с тобой" is definitely incorrect.
Theoretically, you can say "Мы с собакой пошли гулять" (My dog and I went for a walk). In this case, it implies you very close, "human" attitude to your dog. If "мы" = "Я" + "моя собака", then "моя собака" is treated almost as a human being. Otherwise, say just "Я пошёл гулять с собакой".