Can not login from the first try

I'm not sure why that's happening, but I have a few things to try and test out to help narrow down the cause:
* Try enabling the standard mod_login module to see if it has the same issues. That eliminates the SCLogin module and JFBConnect as the problem.
* Disable caching in the Joomla -> Global Configuration area
* Determine why error messages aren't shown in your template. Normally, if you enter incorrect credentials, there should be an error message like "The username or password you entered are incorrect". On your site, there is no error message. That error message would help determine why authentication failed as well. This could be a template setting or the template missing the correct system-messages tag.

Let me know how the above goes and, hopefully, something you find will help narrow down the cause and we can help find a solution.

Thanks,
Alex

1 the standard mod_login gives the same problem;

If the standard Joomla login module has this issue, it means its something other than JFBConnect or the SCLogin module that is causing the problem.

Looking at the URL that the login module is trying to submit to, it seems like it's incorrect. I'd recommend trying to disable whatever SEF extension you're using and see if that makes the issue go away. If so, you may need to contact the SEF provider to determine what the issue is. Again though, from what you tested, it doesn't sound like it's anything directly related to our extensions, and I'm not sure how what we could do to help.

2 caching in the Joomla / Global Configuration area is disabled

Worth a shot. It's always a good thing to test and leave off while debugging.

3 If I use a wrong user/pass I have the message: "Username and password do not match or you do not have an account yet."

Actually, I see that message if I try to login from the login page, but not from the home page. I think that's simply a symptom of the overall login problem and not the problem itself though, so I wouldn't worry about this.

The "missing the correct system-messages tag" is a little bit above me. If you can guide me.

It's not necessary. The message is working on other pages than the home page, and I was incorrect in calling this out as I don't think it's a real issue.

I hope the SEF clue helps narrow things down, but if not, let me know.

Thanks,
Alex