Javascript for dropdown menu (wp_nav_menu) not working

Question

I am attempting to use a script I'm using on another site to get my nav menu drop-down to animate properly.

Here is the script:

jQuery(window).load(function() {

    $("#menu > div > ul > li > a").click(function (e) { // binding onclick
        if ($(this).parent().hasClass('selected')) {
            $("#menu .selected ul").slideUp(100); // hiding popups
            $("#menu .selected").removeClass("selected");
        } else {
            $("#menu .selected ul").slideUp(100); // hiding popups
            $("#menu .selected").removeClass("selected");

            if ($(this).next(".sub-menu").length) {
                $(this).parent().addClass("selected"); // display popup
                $(this).next(".sub-menu").children().slideDown(200);
            }
        }
        e.stopPropagation();
    }); 

    $("html").click(function () { // binding onclick to body
        $("#menu .selected ul").slideUp(100); // hiding popups
        $("#menu .selected").removeClass("selected");
    }); 

});

The html I'm working with is different than that other site I have this working on, as I'm attempting to use Wordpress' default wp_nav_menu() output to make it work. It's a different structure, and I can't figure out what adjustments to make to get it to work properly.

Output of wp_nav_menu:

   <nav id="menu" role="navigation">
    <div class="menu-main-navigation-container"><ul id="menu-main-navigation" class="menu"><li id="menu-item-163" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-163"><a href="#">Programming</a>
    <ul class="sub-menu" style="">
        <li id="menu-item-156" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-156" style=""><a href="http://lgtheater.org/wordpress-2014/programming/bakery/">Bakery</a></li>
        <li id="menu-item-157" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-157" style=""><a href="http://lgtheater.org/wordpress-2014/programming/lg-cabaret/">LG! Cabaret</a></li>
        <li id="menu-item-158" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-158" style=""><a href="http://lgtheater.org/wordpress-2014/programming/mainstage/">Mainstage</a></li>
        <li id="menu-item-155" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-155" style=""><a href="http://lgtheater.org/wordpress-2014/programming/outreach/">Outreach</a></li>
    </ul>
    </li>
    <li id="menu-item-164" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-164"><a href="#">Get Involved</a>
    <ul class="sub-menu">
        <li id="menu-item-161" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-161"><a href="http://lgtheater.org/wordpress-2014/get-involved/audition/">Audition</a></li>
        <li id="menu-item-160" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-160"><a href="http://lgtheater.org/wordpress-2014/get-involved/script-submissions/">Script Submissions</a></li>
        <li id="menu-item-162" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-162"><a href="http://lgtheater.org/wordpress-2014/get-involved/volunteer/">Volunteer</a></li>
    </ul>
    </li>
    <li id="menu-item-167" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-167"><a href="http://lgtheater.org/wordpress-2014/blog/">Blog</a></li>
    <li id="menu-item-165" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-165"><a href="#">About</a>
    <ul class="sub-menu">
        <li id="menu-item-153" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-153"><a href="http://lgtheater.org/wordpress-2014/about/contact/">Contact</a></li>
        <li id="menu-item-151" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-151"><a href="http://lgtheater.org/wordpress-2014/about/history/">History</a></li>
        <li id="menu-item-152" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-152"><a href="http://lgtheater.org/wordpress-2014/about/staff/">Staff</a></li>
    </ul>
    </li>
    <li id="menu-item-149" class="menu-item menu-item-type-post_type menu-item-object-page menu-item-149"><a href="http://lgtheater.org/wordpress-2014/donate/">Donate</a></li>
    </ul></div

>
</div>

EDIT: Adding current menu styles:

#menu {
        float: right;
        position: relative;
        top: 25px;
        }
        #menu > div > ul > li {
            display: inline-block;
            text-align: center;
            padding: 0 1em;
            }
        #menu > div > ul > li.menu-item-has-children a:after {
            content: " \f107";
            font-family: FontAwesome;
            }
            #menu li a {
                color: #fff;
                text-decoration: none;
                font-weight: bold;
                }
            #menu li a:hover {
                color: #F73987;;
                }
            #menu > div > ul > li > ul {
                display: none;
                }
            #menu > div > ul > li > ul.selected {
                display: inline;
                }

Answer 1

In your script $(this).parent() actually refers to li element before the link.

Take the following mark-up for instance

 <nav id="menu" role="navigation">
    <div class="menu-main-navigation-container">
       <ul id="menu-main-navigation" class="menu">
          <li id="menu-item-163" class="menu-item menu-item-type-custom menu-item-object-custom menu-item-has-children menu-item-163">
             <a href="#">Programming</a>

With

$("#menu > div > ul > li > a")

You basically select <a href="#">Programming</a> , thus

$(this).parent().addClass("selected");

adds the class selected to li element, not the ul element which happens to be

<ul class="sub-menu" style="">

right below <a href="#">Programming</a> in your mark-up. You should be adding the class selected to children not to parent.

Try changing the script as follows

$("#menu > div > ul > li > a").click(function (e) { // binding onclick
    if ($(this).next(".sub-menu").hasClass('selected')) {
        $("#menu .selected ul").slideUp(100); // hiding popups
        $("#menu .selected").removeClass("selected");
    } else {
        $("#menu .selected ul").slideUp(100); // hiding popups
        $("#menu .selected").removeClass("selected");
        if ($(this).next(".sub-menu").length) {
            $(this).next(".sub-menu").addClass("selected"); // display popup
            $(this).next(".sub-menu").children().slideDown(200);
        }
    }
    e.stopPropagation();
}); 

EDIT

Not sure how the original menu works, but this may be a solution based on your css definitions.