Is it possible to send messages as a real user (not a page. For example myself) to another user using the Facebook Graph API?
In other words, can I create an app that will make an API call which will send a message as myself to a friend of mine (as if I have sent it myself)
The send and conversation API seems to be only about sending and receiving messages as a page
No, there is only an API for Page to User communication, but not for User to User communication.