Yes, they are grammatically correct. The spelling of "Alright" might raise some eyebrows, but this is a traditionally correct alternative to "All right".
The language is figurative, which is common in songs (and more generally in poetry). I'll assume you have no difficulty in the literal meaning of "I don't exist, if I don't have her." Figuratively, the singer means that being this girl's boyfriend is the most important thing to him: If she is not my girlfriend then things like the sun shining, the world revolving or me existing are nothing to me.
The first and second lines for a complete sentence. The third and fourth lines each have an implied "(If I don't have her) the sun doesn't shine." It is implied by the context.