Good, if you don’t miss any items or metadata, all should be well.
Sorry, there is none. The logs are primarily intended for developers. Issues with recognizing newly added media items should show up first in the scanner logs though (after a library scan has been triggered and finished, that is).
Plex is a bit special when it comes to music.
It definitely requires more preparation of your files and their metadata. Simply pointing Plex to a folder full of unsorted/mixed music files will most likely not get you the result you are expecting.
See Albums appearing with incorrect artist and unable to change it - #2 by OttoKerner for a starter.
Currently you don’t need to expect a music library to show up in the (new) mobile apps. They simply don’t support them yet. Use Plexamp instead.
