Date Picker not working

The reason the calendar icon isn't being displayed for the datepicker is because the datepicker js isn't being loaded, that's also why the datepicker itself isn't working either. You probably have a plugin or theme conflict which is preventing the datepicker js from being loaded. You will need to try disabling and then re-enabling your plugins one at a time to find the culprit. http://www.gravityhelp.com/documentation/page/Testing_for_a_Theme/Plugin_Conflict

Which way are you including jquery, you mentioned you have tried enqueueing it before but that it breaks whatever is using jquery, that is probably because the scripts are being loaded in the wrong order. I had a similar problem with my site but I solved that my enqueueing jquery so it loaded in head while I enqueued other manually added scripts to load in the footer.

I have the following code in my functions file to load jquery in the head

add_action('wp_enqueue_scripts','my_scripts_method');
function my_scripts_method(){
    wp_deregister_script('jquery');
    wp_register_script('jquery','http://ajax.googleapis.com/ajax/libs/jquery/1/jquery.min.js');
    wp_enqueue_script('jquery');
}

This is how I load jquery ui in the footer, it is the 'true' at the end of the call that tells it to load in the footer. I place this in the header file before wp_head

wp_enqueue_script('jquery ui','https://ajax.googleapis.com/ajax/libs/jqueryui/1/jquery-ui.min.js',array('jquery'),false,true);

@SailChick, it looks like that site is using a six month old version of gravity forms, you really should update your installations to v1.6.2