I cant see where in the code you want to have a taxonomy drop-down, and also what you want to do with this drop down. That also depends what values you want to use as option values.
But generally you can use wp_dropdown_categories to create a custom taxonomy drop-down.
wp_dropdown_categories( array(
'show_option_all' => 'Show all arts',
'taxonomy' => 'arts', // Name of the taxonomy or taxonomies to retrieve
'name' => 'arts', // Value for the 'name' attribute of the select element
'orderby' => 'name', // Order drop-down options
'value_field' => 'slug', // Term field that should be used to populate the 'value' attribute of the option elements
// Accepts: 'term_id', 'name', 'slug', 'term_group', 'term_taxonomy_id', 'taxonomy', 'description', 'parent', 'count'
'selected' => $selected, // Value of the option that should be selected. You need to add your own custom $selected
'hierarchical' => true // Whether to traverse the taxonomy hierarchy
) );