The first one is grammatically correct, but it sounds extremely formal. To sound natural in conversation, I would recommend: John has a dog, and Mary has one too. OR John has a dog, and so does Mary.

"John has a dog, as does Mary" is correct. You could also say "John has a dog, so does Mary." Your second sentence is not correct.

Hi Hernandez "John has a dog, as does Mary" is a common way to express that both John and Mary have a dog. "John has a dog, as Mary also has (one)" is also correct, but it sounds a bit more formal and less commonly used in everyday conversation. Hope this helps :)