Is it allowed: kid's music?
Does this mean "childish music" or the music that belongs to the children (like a CD) ?
I believe it refers to music for children.
I think it'd be the former. I'm not sure what the latter would be exactly, but I think that would be "the music of the children," with children being treated as a noun in genitive case. (Музыка детей?)
i think he means referring to music as childish or immature, versus referring to music made for children
Детская is genitive here?
No, it's just an adjective. However, it's an adjective that doesn't have an exact English adjective equivalent. It's this. Or the Russian version.
Children's is the usual way in English to express an adjective "for children". So for example a book written for children to read is a children's book = детская книга
Должен смеяться дети... Должен смеяться дети!