Very sorry for the problems. There's a bug with our forum software and Joomla in general involving UTF-8 usernames. It's something we encounter as an issue rarely, but it's annoying when it happens. Your accounts should be fixed up and properly setup now. You should check your subscriber ID in your "Edit Details" area and make sure it's correct as it may have changed with the fix we made.
As for your issue, the locale you're using (mn_MN) looks correct and Facebook looks to support that. The problem I see right now is that our JFBConnect javascript file cannot be loaded. When I load your page, our Javascript file at the page below isn't loading:
http://www.boditzar.mn/components/com_jfbconnect/includes/jfbconnect.js?v=424
There seems to be something restricting that page from loading. It could be a permissions issue or some sort of security plugin. Whatever it is, you'll need to fix that. Once you do, hopefully everything should work smoothly.
Sorry again for the issues, and please let us know if there's anything else we can help with!
Thanks,
Alex