He's part of a family of four with his mother, father, and dog.
His mother, father, dog, and himself make a family of four.
「He is a family」は間違いです。「彼は家族です」とは言わないでしょう。「He has a family」や「He is in a family」の方が正しいです。
例文２のように、その人が自分の家族に含めてて、再帰代名詞（例えば「himself」）を使った方がいいです。
There are four people in my family: My mother, my father, my dog, and myself.
うちは４人家族です。母、父、犬、と私です。
There are 4 of them, him, his mother and father and their dog
日本語では家にいる人は“家族“になりますね
家族を使うし英語ではliving with （一緒に住んでる）も使える。
彼 - he, him
母と父 - mother and father, mum and dad
1匹の犬 - dog, one dog
４人の家族 - 4 of them, family of 4 , 4 people living together
He lives with his mother, father and dog
There are 4 of them, him, his mother and father and their dog
アメリカ合衆国 
イギリス 
