How can I backup playlists or extract path+filename information from them (generated with builtin MP3-player)?
Playlists are stored in a SQLite database in /data/data/com.android.providers.media/databases/external.db.

I've made a Python script to convert the audio playlist from this database to M3U, it's called m3truid. You need a PC and ADB enabled.

Not as nice as an Android app but especially useful when your screen is dead.