One way is to check whether the user entity has metadata named 'icontime':

if ($user->icontime) {
    // Yes, user does have a profile icon
} else {
    // No, user does not have a profile icon
}

Simplest way is to use jquery bindings:

$(document).on('click', '.link-class', function(e) {

    e.preventDefault();

    elgg.action($(this).attr('href'), {

        success: function(result) {

            // do something when the action is complete

        }

    });

});

There's quite a few steps in order to "ajaxify" things:

1. Register action (or ajax view if there's no sensitive data required)
2. Custom PHP file with necessary Elgg functions and output link/button
3. jQuery click() listener function for custom button
4. Inside the jQuery click function should use elgg.action() function for the ajax request

That's pretty much it I think. See the message board core plugin for an example.

Writing such a plugin would be easy. What would it post?

As I replied on your previous post on the Loginrequired plugin page it would be necessary to know what additional 3rd party plugins you are using the does this form validation on the register page. The reason for the error is very likely due to this plugin (or maybe even plugin if there's more than one involved) not working correctly within a walled-garden Elgg site (neither Elgg core walled-garden or with the Loginrequired plugin). For the plugin to work with walled-garden enabled it would be necessary to allow access to certain pages or actions this plugin provides when not logged in. But it's just impossible to say which one these are without even knowing what plugin(s) are involved.