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?