We don't like to hear that your brains are being eaten.
There's 2 options available:
1) Possibly the easiest way - Make the installers 'upgradeable'
For each zip file you're going to install, unzip it first on your system. Edit the .xml file. In the <install ...> line, add method="upgrade"
So for the JFBCInvite (or any other module), it would look like:
<install version="1.5" type="module" method="upgrade">
Re-zip the file, and then install it. This lets Joomla know that you're ok overwriting any files that are already there.
We'll be using this method in future updates, but for certain reasons, we haven't used it till now.
2) Actually uninstall everything
Delete the files related to each module, plugin, or the main component that's giving issues. This means things like:
/modules/mod_jfbclogin
/plugins/authentication/jfbconnectauth.php
/plugins/authentication/jfbconnectauth.xml
/plugins/authentication/jfbconnectauth/ (the directory)
(do the same for the /plugins/user/jfbconnectuser stuff)
/administrator/components/com_jfbconnect/
/components/com_jfbconnect/
Then, go into your module manager. Make sure you delete any modules. Go to Install/Uninstall, uninstall the component and plugins (the modules shouldn't show up in there at this point). You'll get errors that files aren't found, but this is necessary to clear the database of some things.
Then try to install all over again. Your main JFBConnect component settings should actually still remain after all this. However, you should obviously go through and check each setting.
Hope this helps. Can't say I'm really sure what caused your problems in the first place. It sounds strange. If you have further difficulty, you know where to find us!