I just got into your site to look around. First, I disabled the User - Profile plugin. There were a lot of fields on the registration form, and my guess is you don't need them since you have both Community Builder and Kunena installed on your site. Feel free to re-enable it if you want those extra fields.
With that said, I registered on the site using LinkedIn. My Joomla account was created and a mapping was added to the JFBConnect -> Usermap table. The top menu shows Members | Community | Forum | Logout | Directory | Subscribe, which are correct from what I can tell. My user account is only in the "Registered" group.
That all looks correct to me. I can't recreate what you're saying, and the account is definitely created in Joomla. If there's more information you can provide on how to create the issue.. or, if you want, PM me your LinkedIn login credentials so I can try logging in that way.
I hope that helps, and hopefully you can make some progress on this.
Thanks,
Alex