Does APPLE MUSIC FAMILY PLAN ALLOW EVERYONE TO Share THE SAME LIBRARY?

No. The family plan gives you the ability to share any purchases you've made as well as granting access to Apple Music to the other members of your family. But it won't share your iTunes Match library with them. Even with those purchases, they're required to actually claim those purchases into their own library before using them as part of their library. If your playlists are limited to songs purchased from iTunes and songs acquired from Apple Music, you can share the playlist to mimic the functionality.

Also note that on the AppleTV the iTunes store account is the one used for Apple Music, not any of your Family Sharing accounts. This can make it appear like it isn't syncing playlists, doesn't have a subscription, and other problems.

If the account you use to buy apps isn't the one you use for Apple Music it won't work correctly. This is especially problematic if your iTunes account and Family Sharing accounts are in different regions (which happens a lot with people who married someone from another country, for example).