" Presumably there isn't a way to move rather than copy?"
Nope, the problem is that it must download the message in order to copy it, therefore marking the message as new. It is on the server side that is why it always shows the copies as new.
"Also, can I set up a rule to mark the message as read?"
Sure, you can set it up to automatically mark new messages as read.
"When I tested it the copy showed as unread (though not the original, which is odd)."
Sounds like it is already set up that way