"The name of the Yamashitas dog is Hachi" sounds the best in my opinion.

The Yamashitas' dog is named Hachi. The name of the Yamashitas' dog is Hachi. In the above examples it should NOT be 'Yamashita's dog' because you're talking about an entire family, so there are multiple people. It follows the same rules as any plural. If you say "The Yamashita family's dog is named Hachi", it would be correct.

I would say "The dog in Yamashita's residence is named Hachi."

The name of the dog in the Yamashita house is Hachi, sounds better. You can say "household" instead of house. Or "The dogs name in the Yamashita's household is hachi."