it's a mistake to believe songs are made of grammar textbook content usually quite the opposite because art is far from science

Yeah, the grammar is wrong. Sometimes grammatical errors in songs are due to ignorance or carelessness (I recall one song from the 1970s that incorrectly had "could of" instead of "could've"). However, in the case of the Beatles, it probably was a deliberate choice for smoother lyrics ("doesn't" is kind of an awkward word).

You're correct. In informal situations, and in songs (especially rap music), sometimes grammar rules are broken/relaxed to suit the speaker.

You are correct from a pure grammar point of view. In spoken English you might say she don't. In the song she doesn't wouldn't work.