All of them are right. We can say "he has a dog" " Has he a dog?" " Has he got a dog?" or " Does he have a dog?". The difference is that "have got" is used in spoken English and "have" is used in both spoken and written English. Question "Has he a dog?" is correct, but as far as I know, it is obsolete.

This is from a British English perspective: “Have got” and “have” mean the same. “Have got” is more informal. If you want to ask a question you can ask: “Has he got a dog?” Or Does he have a dog?”

Hello, In everyday British English, people usually say have got something rather than have something, but in writing they usually prefer to use just have: ■ He’s got a degree from Bristol University. (spoken, everyday) ■ He has all the relevant qualifications. (written)