Question about Japanese culture?

[Surgical With Chopsticks]

I know they use their surnames as their first names and go by their last names, usually. For example, I see Japanese baseball players with their first names on the backs of their jerseys. If a woman from Japan marries an American man, which name do she keep? Does she keep her first or last name? Or does her last name become her first name like the Vietnamese do? My other question about Japanese culture entails animals. Do they have dogs or cats as domestic pets there? Or do they have other animals like rodents or birds?