Glad to hear you got the login buttons to show.
As for the profile pic as a drop-down menu selector, I'm not sure of the best way or how to do this. If you've copied the styles and HTML, what is different? I'd open the 'old' site and use the Developer Tools in your favorite browser to see where the HTML differs.. and narrow it down from there. I don't have any specific suggestions though as it's not something we've done before.
Can you also explain why after logging in, a registered user gets an error message at the top of the page saying "Error. You are not authorised to view this resource."
Whatever URL the user is on has permissions that are higher than their pay grade

I'd look at the Joomla menu manager or in the extension of the current page to determine what resource is being loaded. Also, look at the Login Redirection setting in JFBConnect to make sure they're going to the right page.
Thanks,
Alex