I see there are many 3rd party clients out there that support FB chat, but are there any that will show the previous messages in the chat, from before the conversation was started in the 3rd part client (similar to the way the FB Messages iPhone app works)?
So far I have checked out Adium and Trillion (OSX vers only) and they only show the messages sent in the current conversation.
In short, does the public API have enough support to build a chat app that will allow offline messaging?
No, but you can use REST API call /me/inbox to retrieve history, we are using it on ChattAR for Facebook sample app, it's open project