As for putting the invite module (or any module) into your user's profile page, you can use the following code. Hopefully this helps overcome the issue. We have some mods to the javascript which we'll be getting out very shortly which may help the issue, but it seems that some people's sites operate differently, so it's not guaranteed it'd work for you. Hopefully, the below will do exactly what your looking for:
Lets say we're going to call our new module-position "jomsocialinvite". First, create a new module in the backend and assign it to this position (just type "jomsocialinvite" into the Position field). Then, add this code to your template file wherever you'd want the Invite module to appear. A possible file to add this to would be 'profile.header.php':
$module_position = "jomsocialinvite";
$attribs = array("style" => "rounded");
$subModules = &JModuleHelper::getModules($module_position);
foreach($subModules as $subModule)
{
JModuleHelper::renderModule($subModule, $attribs);
print $subModule->content;
}
You could then place as many standard Joomla modules in your profile as you want.