Is it possible for every band member to have it's own lyrics? I (guitar) have often different chords than the keyboarder. If it's possible, how does it work? I couldn't find it out...
Yes, users can turn on Settings > Account Sync > Personal Sync > Lyrics, and then they will see two lyrics fields on the song edit page -- one to enter their own personal lyrics that nobody else sees, and one for the shared lyrics that everyone sees. Then the app will display the personal lyrics if they exist for a song, or the shared lyrics if there are no personal lyrics for that song.
Which fileds of a song are "private" to a member (so that every member could have it's own values in it)?
You can look on Settings > Account Sync > Personal Sync for other fields that have a personal option. Additionally, custom fields, documents, recordings and MIDI presets can be assigned to one or more specific users, and then only those users will see them.
"Private" songbooks can be realized using the project feature and assign only one user to it?
Yes:
http://www.bandhelper.com/tutorials/projects.html