Should I use the authentication token returned from the Facebook login as the password to the web service?
I recommend using one of our supported SDK's to handle the login auth. You, as the developer, does not have to worry about storing passwords/tokens/etc. By using fb login, we handle all of that. I recommend reading more about using Facebook as a login here.
I'm not sure on where the token is stored and how to retrieve it? Also how could I update the web service with the new token when it expires?
I would recommend reading this doc to see how we recommend to retrieve and update access token when it expires.