Is it correct to write and dog?

No, "and dog" is not a complete phrase or sentence. It needs more context to be grammatically correct.

Here are some ways to use "and dog" correctly:

* As part of a list: "The children were playing with the cat and dog."

* As a conjunction: "I went to the park with my friend and dog."

* In a sentence describing an action: "The boy threw the ball, and the dog chased it."

It's important to remember that "and" is a conjunction, and it needs to connect two or more elements (words, phrases, or clauses) to make a complete sentence.