I checked the mail functionality with 6.0.3 Notes Clients… and the firt issue I noticed is that in the local Replica, no read marks are passed, regardles I configured them in the advanced properties:
I would get rid of the local replicas completely, get the unread marks where you want them on the server copy (or copies), and then pull down new local replicas. Unread marks are difficult to deal with, and it can be easier to just start fresh sometimes!