Appcelerator Cloud Chat API

1. I am looking for a solution to build a Chat Service between a WebApp(Desktop) and Mobile App. Does Chats API support this model?
2. What is the underlying implementation of Chat service in cloud? is it XMPP and PushNotification combination? Are there any Apps that are built on this API?
3. What happens when the device is off and other users send messages? Are the messages synced when the device is ON.
4. What happens to the messages on iOS when the App is not running?
5. If Chat API does NOT support this model (Web <--> Mobile) , What do you suggest XMPP server/client or socket.io or WebSockets?

Really appreciate your answers.

Thanks Suresh Velagaleti