VOST

Thanks bcworkz in trying to help me out; it took me some times but I eventually found something.
What follows is not a long-term solution. If you wish to keep in your database the fact that you assign terms of custom taxonomies to a certain post using repeatable fields, then, I suppose, it evolves doing a bit a SQL.
Still, I am posting my solution as it may be of use for someone.
The trick, compared to my previous code, was to add a count variable (here $cnt) to the name attribute of my select element:
Before:
$html .= '<select name="screeningName" id="screeningName">';
After:
$html .= '<select name="screeningDetails[' . $cnt . '][screeningName]">';
Semantically, the repeatable fields are tr elements of a table, with the remove button within each row. In order to make it work (the remove button inside each row) the jquery code needs to be changed:

$(".remove").live('click', function() {
        //$(this).parent().remove(); // When the button is outside the repeatable element
        $(this).parents('[class*="repeatableField"]').remove(); // When the button is inside the repeatable element
});

So, here we go, follows the corrected code (I also pastebin-ed it for anyone to improve it, https://pastebin.com/CxWaiqQk):
Drop-down list:

function print_screening_field($cnt, $p = null) {
if ($p === null) {
    $a = $b = $c = '';
} else {
	$a = $p['screeningName']; // Change variable name
	$b = $p['screeningMonth']; // Change variable name
	$c = $p['screeningYear']; // Change variable name
}
	$festivalTerms = get_terms('festival', 'hide_empty=0'); // Change variable name and taxonomy name
	$festivalNames = wp_get_object_terms($post->ID, 'festival'); // Change variable name and taxonomy name
	$monthTerms = get_terms('month', 'hide_empty=0'); // Change variable name and taxonomy name
	$monthNames = wp_get_object_terms($post->ID, 'month'); // Change variable name and taxonomy name
	$yearTerms = get_terms('year', 'hide_empty=0'); // Change variable name and taxonomy name
	$yearNames = wp_get_object_terms($post->ID, 'year'); // Change variable name and taxonomy name

	$html .= '<tr id="'. $cnt .'" class="repeatableField form-field">'; // The class name here is important regarding the jquery's part of the code
    	$html .= '<td><label>#'.$cnt.'</label></td>'; // Here the $cnt variable is used just to visually see how many fields we have created

    	$html .= '<td>';
			$html .= '<select name="screeningDetails[' . $cnt . '][screeningName]" id="' . $cnt . '">'; // to name the select makes it work, adding also the $cnt variable to it
				$html .= '<option value="';
					if (!count( $festivalNames )) {
						$html .= 'selected">';
					}
				$html .= 'Select a festival</option>';
				foreach ($festivalTerms as $festivalTerm) {
				$html .= '<option value="' . $festivalTerm->name . '" '.($a==$festivalTerm->name?'selected':'').'>' . $festivalTerm->name . '</option>';
				}
			$html .= '</select>';
		$html .= '</td>';

    	$html .= '<td>';
			$html .= '<select name="screeningDetails[' . $cnt . '][screeningMonth]" id="' . $cnt . '">'; // to name the select makes it work, adding also the $cnt variable to it
				$html .= '<option value="';
					if (!count( $monthNames )) {
						$html .= 'selected">';
					}
				$html .= 'Select a month</option>';
				foreach ($monthTerms as $monthTerm) {
				$html .= '<option value="' . $monthTerm->name . '" '.($b==$monthTerm->name?'selected':'').' id="' . $cnt . '">' . $monthTerm->name . '</option>';
				}
			$html .= '</select>';
		$html .= '</td>';

    	$html .= '<td>';
			$html .= '<select name="screeningDetails[' . $cnt . '][screeningYear]" id="' . $cnt . '">'; // to name the select makes it work, adding also the $cnt variable to it
				$html .= '<option value="';
					if (!count( $yearNames )) {
						$html .= 'selected">';
					}
				$html .= 'Select a year</option>';
				foreach ($yearTerms as $yearTerm) {
					$html .= '<option value="' . $yearTerm->name . '" '.($c==$yearTerm->name?'selected':'').'>' . $yearTerm->name . '</option>';
				}
			$html .= '</select>';
		$html .= '</td>';	

		$html .= '<td>';
			$html .= '<button class="remove">Remove #' . $cnt . '</button>';
		$html .= '</td>';

	$html .= '</tr>';

return $html;
}

Display on the admin side:

function screening_details_meta_box(){
	global $post;
	$custom = get_post_custom($post->ID);
  	$screeningDetailsData = get_post_meta($post->ID,"screeningDetails",true);
?>
  	<h4 style=" text-transform: uppercase;">Other screenings</h4>
  	<table id="data_items" class="form-table" border-collapse="collapse">
		<caption style="display: none;">List of screenings other than the premiere screenings</caption>

		<thead>
			<tr class="form-field">
				<th>Number</th>
				<th>Name</th>
				<th>Month</th>
				<th>Year</th>
				<th></th>
			</tr>
		</thead>

		<tbody>
  		<?php
 	 	$c = 0;
    		if (count($screeningDetailsData) > 0){
    		    foreach((array)$screeningDetailsData as $p ){
    		        if ( isset($p['screeningName']) || isset($p['screeningMonth']) || isset($p['screeningYear']) ){
    		            echo print_screening_field($c,$p); // uses the fonction name
    		            $c = $c +1;
    		        }
    		    }
	    	}
		?>
		</tbody>
	</table>
    <span id="here"></span>
    <button class="add"><?php echo __('Add a screening'); ?></button>
    <script>
		var $ =jQuery.noConflict();
		$(document).ready(function() {
		var count = <?php echo $c; ?>;
    	$(".add").click(function() {
			count = count + 1;
        		$('#data_items').append('<? echo implode('',explode("\n",print_screening_field('count'))); ?>'.replace(/count/g, count));
        			return false;
        		});
        		$(".remove").live('click', function() {
        			//$(this).parent().remove(); // When the button is outside the repeatable element
        			$(this).parents('[class*="repeatableField"]').remove(); // When the button is inside the repeatable element
        		});
        	});
    </script>
    <style>
    	table#data_items { margin-bottom: 20px; }
    	tr[class*="repeatableField"]:nth-child(odd) { background-color: #EEEEEE; }
    </style>
    <?php
}

Save the data:

function save_meta_box($post_id){
	global $post;

	if ( defined('DOING_AUTOSAVE') && DOING_AUTOSAVE )
	return $post_id;
	if (isset($_POST['screeningDetails'])){ $screeningDetailsData = $_POST['screeningDetails']; update_post_meta($post_id,'screeningDetails',$screeningDetailsData); } else { delete_post_meta($post_id,'screeningDetails'); }
}
add_action('save_post', 'save_meta_box');

Hello bcworkz,

thank you for your link. What I am trying to do is slightly more complex as I wish to have several drop-down lists (so several values) embedded in one repeatable field. Visually it would be something like such:

[repeatable]
– Drop-down: Screenings’ names
– Drop-down: Screenings’ countries
-remove button
[/repeatable]
[Add a screening button]

So far I succeeded to return the drop-down lists (which was my first problem) but I still cannot find a way to attach values to the selected options.

Here is my code so far (what changes, compared to the first code in my first post, is that I closed my condition “$p === null” before I set my $festivalterms and $countryterms variables):

function print_screening_field($cnt, $p = null) {
if ($p === null) {

    $a = $b = '';

} else {

    $a = $p['screeningName'];
    $b = $p['screeningCountry'];

}

    $festivalterms = get_terms('festival', 'hide_empty=0');
    $festivalnames = wp_get_object_terms($post->ID, 'festival');
    $countryterms = get_terms('country', 'hide_empty=0');
    $countrynames = wp_get_object_terms($post->ID, 'country');

	$html = '';
        $html .= '<li>';

	$html .= '<label>Festival: </label>';
	$html .= '<select name="screeningName" id="screeningName">';
	$html .= '<option value="">Select screening</option>';
	foreach ($festivalterms as $term) {
		$html .= '<option value="' . $a . '" ' . selected($a, '' . $term->name . '') . '>' . $term->name . '</option>';
	}
	$html .= '</select>';

	$html .= '<label>Country: </label>';
	$html .= '<select name="screeningCountry" id="screeningCountry">';
	$html .= '<option value="">Select country</option>';
	foreach ($countryterms as $term) {
		$html .= '<option value="' . $b . '" ' . selected($b, '' . $term->name . '') . '>' . $term->name . '</option>';
	}
	$html .= '</select>';

	$html .= '<button class="remove">Remove</button>';
	$html .= '</li>';

return $html;
}