You should have the Facebook JomSocial Connect integration disabled completely. To do that, simply remove the API/Secret key from their settings and that should do it. Feel free to put the other options to 'no', but I think they are ignored when the API key is blank. Looking at your site, the keys seem to be blank.
I did just click the Login With Facebook on your site and was brought to the following page:
index.php/component/jfbconnect/?view=loginregister&return=L2luZGV4LnBocC9jaGVlci1uZXR3b3Jr
Once there, I was seeing additional JomSocial fields to be filled out. I filled them out and my info was saved along with my avatar. One thing I did notice during registration is that it looks like you are using Kunena v2.0 and have the Kunena/JFBConnect profile plugin enabled. We haven't upgraded our profile plugin to support Kunena v2.0 yet (it just came out about a week ago). Right now, we'd recommend disabling that plugin. That may be part of your issues with logging in and updating the avatar, just so you know.
We're hoping to have a Kunena v2.0 compatible plugin out later this month.
Hope that all helps, but keep us posted on what you find! Also, feel free to delete my user/profile from the site.
Alex