When I delete a song from my library , it is removed from the playlist

No.

Music, (like iPhoto, Aperture, Photos, iTunes) is a database. This means it can be very flexible and powerful, if sometimes a little confusing. The library is the core of the app. Every track is in the library. playlists are virtual, and are just subsets of the library. In fact, playlists are not places, they are lists - instructions to the app to 'play this group of tracks together'. So tracks do not get "moved" to playlists, they get added to that particular list. This means that an image can be in more than one list but use no extra disk space.

Nope. That is the expected behavior and is consistent with how the Library works in other apps. If you remove a Photo from your Library, it is also removed from an album that contains it.