I think this issue is caused by a combination of Windows Libraries sharing permissions & a situation with Windows network sharing permissions where discrete paths to and from offline files each also have their own discrete permissions. I'm not an expert on either, but I did manage to get it working.
In my case, I was redirecting the Documents shell folder (on a Win7 computer/client) to a network location (Win10 computer/server) like you...which automatically set up the Offline Files sync partnership. It's worth noting that this automatic Offline Files sync partnership shares the entire User folder on the server...not just the Documents folder even though the Documents folder is the only one that is being redirected...and the Documents folder is the only folder that shows as Always available offline on the client. In this case I don't think Windows sets up all the permissions correctly --- Offline Files expects the files on the client computer to be the 'master' files that are mirrored back to the server, but in my case I wanted all the folders & files on my server to be the 'master' files that get mirrored to the client...in essence using Offline Files as an easy way to set up a fileserver with automatic/background client sync. So my client computer essentially becomes a 'dumb terminal' where I can edit files & directories (whether I'm connected to the server or not) and all the changes get mirrored back to the 'master' files on the server when I'm back on the same network as the server. I don't think this is the intended use of Offline Files...but it works.
When viewing my offline files on the client (through the Sync Center/Manage offline files/View your offline files interface) I witnessed the same behavior as you --- My Music / My Videos / My Pictures folders unexpectedly appearing inside the Documents folder...which do not exist on the server...and are greyed out...and throwing access denied errors when syncing. This is what clued me in to look at Libraries.
My strategy to fix the access denied sync issue with the My Music / My Videos / My Pictures folders that automagically (erroneously? confoundingly?) appeared in the synced Documents folder on the client was to remove and then re-add the Documents folder to the Documents Library (right-click/Properties) on the server. The Documents folder icon in the Documents Library Properties page then reflected that the server-side Documents folder was shared. This seemed relevant...and made me hopeful...but didn't entirely fix the issue.
The next step was to go back to the client computer and browse to the network location of the redirected Documents folder...through the full network path (Network\Computer\Users\User\Documents)...and make sure Always available offline was checked. I think this automagically set the correct permissions that allowed Offline Files to properly process and sync the mystery Library folders (My Music / My Videos / My Pictures). In other words, I don't think Offline Files set the permissions correctly for the entirety of the network path from the server to the client because it only expects to use the network path from the client to the server. If that makes sense because, again, I think this is a non-standard use of Offline Files.
Anyway, it was great to see Sync Started and sync Completed in Sync Center after I got this set up properly.
Hope it helps!
<<EDIT This probably won't help --- please see my definitive explanation and solution in the comment below...>>