Alright, I'll answer what I can.. if I leave something out, just let me know

The FB login box will ask for your credentials if you're not already logged into Facebook. If you are logged in and click the blue button, it will pop-up immediately and then go away (may not even see it). IF it's the very first time you've tried to log in, it will ask for your permissions. If you've already granted them, you won't get the permission dialog box.
Now, why it's not logging you in and redirecting you, that's a problem. I just tried logging into your page, and on the redirect from Facebook, I'm getting the following Javascript error:
Error: document.getElementById("jfb_nav_menu_wrapper") is null
Source File:
www.wingchun-kungfu.co.uk/index.php
Line: 127
Do you know what may be causing this? The Facebook login process is very Javascript dependent, and errors can prevent the proper 'flow'. I think if you can get that worked out, the login stuff should go smoothly.
The blurry avatar is a bug in our code. We'll be releasing an updated version of JFBConnect this weekend (the stable version) which will import a more crisp avatar into JomSocial. Sorry about that.
We don't have a method to prevent associations with admin accounts. However, in the Login Module, there is a 'Allow Reconnections' parameter. You can turn this off to prevent an account that is already linked to be linked to a new account. To 'unlink' an account manually, in the JFBConnect Admin area, click the "User Map" and you can delete the row which maps the Facebook user to the Joomla user. We have a task to investigate better methods of this, but it won't be out until after 3.0 stable. Same with preventing administrator linking.
I'm not sure what logout button your talking about specifically because you say the JomSocial module both times. However, the reason you're being logged back in is because you likely have the "Automatically Login Facebook Users" setting as "yes". When you use the JomSocial logout button, you're logging out of Joomla and not Facebook. On the next page load, JFBConnect 'sees' your FB user and tries to log them back in. In short, if you're not using the JFBConnect Login Module, set the "Automatically Login Facebook Users" to 'No'. We have a note about this problem in the next version to warn admins about the possibility of never being able to log out. I'm not sure where the OAuth error is coming from. Feel free to report it again if you continue to see it after the above.
Whew. Hope that helps, and let us know if you have more questions. We'll get you going!