Ivan,
I understand that the issue that was broke is working now.. but now we've introduced another issue. To fix this, it's possible something else may break, which is why I'm saying that maybe we tackled the initial issue (the tabs) incorrectly, and possibly we should look for other solutions to the original problem instead. I have a suggestion below for the current issue, but with each change, we're going farther and farther away from what Facebook recommends, which means that other issues (with Facebook or other parts of your page) may likely crop up due to these changes.
With that said, when the page refreshes and goes white like that, it's usually a problem with the channel URL setting for Facebook. We used to not use this setting, and the white page problem was more common. We've never heard of it since we added the setting, but in your case, you may want to try removing it. In the JFBCSystem plugin, simply remove the ", channelUrl: '{$channelUrl}' value. Again, that should be there for proper Facebook operation, but it can work without it.
If that doesn't work, then we'd need to investigate the Javascript errors that are appearing in IE: this.$events is null or not an object in mootools.js. Again, JFBConnect doesn't use Mootools on every page, so this isn't coming from us directly, and I still don't know what's causing a conflict on your site or why. If you know more about what's using Mootools on your site, that would help us debug this issue.
Thanks,
Alex