It would be easier if you had any explicit questions. But I'll try.

What looks wrong to me are the big diagonal two-way arrows from "My Library" to what you call "Group Library Mother Ship".
Group Libraries do not sync with personal libraries (My Library). They sync with local copies of the Group Library --> The horizontal arrows from Group Library Mothership to Local Group Library that you have (though those should be two-way arrows as well).

On the other hand, you might want to have two-way arrow between My Library and Group Library at the local level, where you can drag items (whether notes, links and attachments are dragged is an option in the Zotero preferences. Attachments further depend on the group's settings).
None of this is "synced" or automatic, though.

All syncing includes conflict resolution. All syncing includes tags and notes.
Syncing in Private Groups can include attachments (when allowed by the group owner, using Zotero storage, and if the group owner has enough storage volume.

that looks good. The syncing between the local and cloud version of "My Library" is two-ways, too (e.g. if you sync to the cloud from two different computers, like your laptop and office PC), i.e. the arrows that you have labeled with "Auto Option" go both ways, too.

As for your questions, there just seem to be problems with syncing not working correctly for you:
http://www.zotero.org/support/kb/changes_not_syncing


For the questions on your figure -
Note that you turn auto-sync on or off entirely - if your library auto-syncs so do groups.

I don't know if I understand what you mean by "conflict btw. local groups" - if two people in a group edit the same item you'll get conflict resolution.