"He has a dog?" should be accepted too, no?
I got this one wrong as I would say "has he a dog?" but I guess this is probably Glaswegian dialect and not proper English.
It is proper English, only a bit old-fashioned for Americans. They should accept it, so you should report it when you get it again.
"Does he have a dog?" is more proper and does sound more common in English.
"Has he a dog?" is proper English and should be accepted.
Seems a bit as if the structure of the sentences is similar to German :)