Wp Nav Menu div containers

From what you described, it sounds like you might need to use a custom walker for the wp_nav_menu call. To create a custom walker that modifies the wrapper for dropdown lists, start with this class:

<?php

class wpse_112127_walker extends Walker_Nav_Menu {

    function start_lvl( &$output, $depth ) {
        $indent = str_repeat("\t", $depth);
        $output .= "\n" . $indent . '<div class="dropdown"><ul>' . "\n";
    }

    function end_lvl( &$output, $depth = 0, $args = array() ) {
        $indent = str_repeat("\t", $depth);
        $output .= "$indent</ul></div>\n";
    }
}

Then, use your walker in your wp_nav_menu call:

wp_nav_menu( array( 'walker' => new wpse_112127_walker ) );

This code should hopefully get you started on using a custom walker for your use case. Let me know if you have any difficulties/questions!