Carlos,
Sorry for the delay. Your Facebook application is likely setup fine, since login and logout is working, even though it's not working all the time. It wouldn't work at all if your application was incorrect. Hwoever, in the 4.2 release (out tomorrow), there will be an Application checker tool built-in to check for common problems and give recommendations for how your app should be setup.
The issue is definitely a caching problem. I'm testing in Chrome right now, and am stuck in a situation where I can't logout of the site. Each time the page redirects me to the home page, but the cached version is being loaded. The cached version is of the page with me logged in, it looks like. The logout link itself is a bit strange since it goes to: index.php?Itemid=98. Not sure why it looks like that. May be normal, but it's not the normal link.
However, after I typed that, I tried the logout link int he SCLogin module that drops down, and that worked fine. So, not sure if it's just that link in the right sidebar, but something is strange there as well. This was in Chrome.
Anyways, going back to the original login issue, the problem is definitely an agressive browser caching scheme that is setup for your site. If you look at the image below, you'll see that the request made to our LoginFacebookUser page is simply being returned from cache:
The top row is from right after clicking the Login With Facebook button. The "size" column shows that chrome is simply loading the resulting page from it's own cache instead of actually trying to send the request and execute the page.
That would explain why changing the URL fixes the issue. Since that seems to work for you, feel free to try the following change, which will make the login URL random every time a user clicks it. In the /components/com_jfbconnect/includes/jfbvconnect.js file, at line 29, you'll see:
self.location = jfbcBase + 'index.php?option=com_jfbconnect&task=loginFacebookUser&return=' + jfbcReturnUrl;
Update that line to be:
var d = new Date();
self.location = jfbcBase + 'index.php?option=com_jfbconnect&task=loginFacebookUser&return=' + jfbcReturnUrl+'&cb='+d.getTime();
You'll need to clear Joomla and browser cache when done. Hopefully, your CDN won't have the jfbconnect.js file cached either (or you'll need to clear it or wait for it to expire).
Let us know how that goes, and good luck,
Alex