problem with login with FB

1. the system showed me mistake (please find attached the screenshot),

There was no attachment in the post. After you upload it, please make sure you click the green arrow to insert it into the post. Without seeing the message/mistake, or a description, I'm not sure what is happening.

2. system said that I have to activate my account again with link sent through e-mail.

Are these for users that register with a social network using JFBConnect, or from standard users that register or are logging in through Joomla with their username & password? If it's from a social network, can you tell me your setting for "Skip Joomla Activation"? If you have activation enabled, then it would make sense that the user needs to go through the activation process.

3. After re-registration some modules looks strange, buttons are in English again, instead of Russian.

How are you re-registering? How did you delete the user (in K2 or in the Joomla manager)? What buttons are in English? Does this problem ever happen with other users that you haven't deleted?

Sorry for all the questions, but I'm really unsure as to what's happening on your site, which makes it difficult to diagnose. Any information you can provide will greatly help.

Thanks,
Alex

They are in k2 list users (not in the list of users with FB account). But they are registered with SCLogin form as I have only this form for registration in my site. And every day appears at least 5 new weird users. It looks like virus or spam. Could you help me please!

It very likely is spam. There's nothing the SCLogin module can do to prevent this though as we use the standard Joomla registration process. Automated tools are out there to scan any Joomla site and find the registration page, which on your site is:
2012-12-13-08-52-53/joomla-user/user-registration
or
index.php?option=com_users&view=registration (This is the standard registration page)
There's no captcha or any checks on that page from preventing users from creating any account they want. I'd recommend any (or all) of the following:
* Turn on Captcha in Joomla
* Make sure email activation is enabled. You can skip email activation for accounts created through JFBConnect
* Install a spam registration extension like SpamBotCheck, which we use on this site:
extensions.joomla.org/extension/spambotcheck

I hope that helps,
Alex

As far as I know it is not possible to use captcha in SCLogin form

That's correct. The SCLogin module *only* provides a means to log the user in. Logging in shouldn't need captcha and should only be used during registration.

The registration page itself it handled by Joomla or whatever 3rd party extension you may be using for registration. When you click the "Register" button in the SCLogin module, it will simply redirect there. The SCLogin module doesn't have any control over the registration process at that point other than redirecting the user to the registration page.

I hope that helps explain a bit more, but if you still have any questions, just let us know.

Thanks,
Alex