Search box with field hidden till drop down selection is made

To add the data-select to the select tag use

 <?php 
   $select = wp_dropdown_categories($dropdown_args); 
   $select = preg_replace("#<select([^>]*)>#", "<select$1 data-select="select1">", $select); 
   echo $select; 
 ?>

then change the echo in $dropdown_args to 0.

I will take the jquery part of the question to the right platform