JFBCFan tags appear in frontend instead of the expected box

That looks fine. Could you PM me superadmin access to your backend? I might try to install JFBConnect 6.3 on your site (it will be released later this week anyways). There are a couple of bugs that could affect this which are fixed in the release.

(of course, we also noticed that the basic functionality, that of our visitors being able to login, also does not work... which is more serious. But we guess that it is the same problem? The js not loading).

No, there is another extension on your page that is adding the deprecated all.js Facebook Javascript Library. This is an older, deprecated version of the library that Facebook has not recommended for use in over a year. JFBConnect uses the recommended sdk.js library. If you want the Facebook login functionality on your page, you will have to remove the older library from being included as 1) the two libraries won't work together and 2) coming in April all.js will not support login functionality anymore (which is why it is no longer recommended). The following is the code that is hardcoded/needs to be removed.

This is different from the issue that I am looking at now. I have your admin credentials, but could you also PM me FTP credentials as well? Installing 6.3 did not fix the issue so I will need to do some more in-depth debugging efforts.

Just wanted to note that I found where the all.js was hardcoded in your template index.php and I have removed it. By the way, I am keeping a list of steps I've taken, so it will be easy to migrate this live.

I was able to get the Fan box loaded and able to create a user and authenticate with my Facebook credentials.

1. I installed JFBConnect v6.3 on your site. This didn't seem to affect anything to solve the problems you were having.
2. I removed the all.js inclusion that was hardcoded into your sintagepareas index.php file.
3. I disabled the 'System - Core Design Scriptegrator' plugin. When this is enabled, there is some conflict that prevents our system plugin from working properly. Our javascript files/CSS files etc don't get added to the page properly. Simply re-ordering the plugins does not work. I don't know if this plugin interferes with any other extensions on your system, but you will need to investigate this further.
4. After disabling the plugin, the fan box loads properly.
5. After #2 and #3, clicking on the Facebook button to create a user looks like it starts to work. However, once you hit the register button on our login / register page, it goes to a blank page (meaning there is a PHP error). I then turned on error reporting to Development mode and the error is not from JFBConnect.6. Even though there was a blank page with an error, JFBConnect has created the user properly. You can log in subsequently using your Facebook credentials.

Note
* I have switched error reporting back to Simple, since you have a lot of warnings on your page.
* I have left my "Melissa Lynne SourceCoast" user on your site, so that you can see that it was mapped properly in JFBConnect > User Map table, but you can delete this.
* I have left the 'Core Design Scriptegrator' plugin disabled.

From the above steps, I don't think that I will need FTP access anymore. You will need to investigate 1) the Scriptegrator plugin as this seemed to be causing the core problem for Facebook javascript and css not being included properly 2) the com_uddeim extensions which is throwing fatal errors and 3) there are a lot of warnings on your page that I'd recommend trying to resolve.

-Melissa

Not a problem. Let us know if you run into any other oddities or strange behavior that might be related to JFBConnect.

-Melissa