Dropdown list of taxonomies won’t display selected

It seems you are adding the term ‘selected’ to each option – you only want it added to the currently chosen option.

You can use the WordPress built-in function, selected, which allows you to compare two values (say the saved term’s ID and the current options term ID) and when they match prints ‘selected’.

For instance:

<?php
$current_term;//Retrieve the current saved term's;
//Get the current term's ID, or 0 if it doesn't exist
$current_term_id = ($current_term ? $current_term->term_id : 0); 
?>

<select name="meta-box-events_select" id='meta-box-events_select'>
 <option value="" <?php selected(0,$current_term_id)?>>Select Term</option>
        <?php 
        foreach ( $taxterms as $term ) { 
            echo '<option value="'.$term->slug.'"'. selected($term->term_id,$current_term->term_id).'>'. $term->name . '</option>'; 
        } ?>
 </select>