I didn't have the Mail->Preferences->Viewing "Mark all messages as read when opening a conversation" checked. So I guess that wasn't the problem.
I have checked "Display unread messages in a bold font" and "Display remote images in HTML messages" boxes and neither actually happens. I've tried unchecking them and shutting down the computer, then starting up again and checking them and shutting down again etc to no effect. I've done this several times over the last six months and the email account has never displayed the way it is meant to.
I have mail accounts for the MacBook Pro (which has OSX Mountain Lion 10.8.5) for my iPad3, my iPhone4 and an old MacBook which the kids use. It's only the MacBook Pro which has the problem. So I presume that the junk filtres used by my email provider is not causing the problem, as it is only one device that it happens.
I went onto my old MacBook (which is Snow Leopard) looking for an old email today and noticed that the email displayed properly and it made me realize how irritating it is that the email show as read when they are not. How confusing even.