"She has no dog."
Translation:Hon har ingen hund.
inte någon, or with a very informal spelling, inte nån means 'not any' in English. Now, in English it isn't idiomatic to say "she doesn't have any dog", so you'd just say a dog instead, but in Swedish, hon har inte någon hund is perfectly natural. In the plural, it works the same again: hon har inte några hundar is she doesn't have any dogs.
This sentence reminds me of the poor English spoken in more rural areas of the country. "She has no dog" is very bad English and doesn't even sound good, let alone correct. In English it would be "She doesn't have a dog." Hmn .... wondering if it could be stated in Swedish as: "Hon har inte en hund."