Howdy,
Both issues are almost always caused by the ordering of the user plugin. If you look at the JFBConnect Overview page, in the "Required Extensions" area, there is likely a warning showing up there. Please ensure the JFBConnect - User plugin is ordered higher (and not equal) to the Joomla user plugin. Once a user registers through Facebook/JFBConnect, the "Connect this account" button shouldn't appear.
If this doesn't help, let us know, and we'll help you figure it out!