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.

I agree with what Jonathan Kimball wrote. The first sounds natural but too formal for everyday conversation. It would be fine in writing that wasn't meant to be conversational. The second doesn't sound natural at all. And Jonathan K's recommended conversational alternatives are what I would have recommended as well.

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 :)