Hi,
Thank you for your reply, which helped a lot. (I do have some questions below)
This is the result that I came up with.
Maybe it can still be made easier, so anyone feel free to make it cleaner and shorter. (I don't mind)
Oh yes, I added a if-statement to be able to select(selected="selected") the right value ($row->cart_product_quantity).
<select id="hikashop_checkout_quantity_<?php echo $row->cart_product_id; ?>" name="item[<?php echo $row->cart_product_id;?>]" class="hikashop_product_quantity_field_dropdown">
                <?php
                $min_quantity = $row->product_min_per_order;
                $cart_product_quantity = $row->cart_product_quantity;
                $jmax = $min_quantity*10;
                for($j=1; $j<=20; $j++) {
                  $multiple_min_quantity = $min_quantity * $j;
                  
                  if ($multiple_min_quantity == $cart_product_quantity) {
                  ?>
                  <option value="<?php echo $multiple_min_quantity; ?>" selected="selected" onchange="var qty_field = document.getElementById('hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>'); if (qty_field){<?php echo $input; ?>}"> <?php echo $multiple_min_quantity; ?></option>
                  <?php }
                  else {
                  
                  ?>
                  <option value="<?php echo $multiple_min_quantity; ?>" onchange="var qty_field = document.getElementById('hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>'); if (qty_field){<?php echo $input; ?>}"> <?php echo $multiple_min_quantity; ?></option>
                  <?php
                  }
                }
                  
                ?>
                </select>
I changed the class name of the select into 
.hikashop_product_quantity_field_dropdown and added the following in the 
frontend_custom.css.hikashop_product_quantity_field_dropdown{
	float:left;
}
1) Does the onchange in the option value do something. I'm not sure what $input is used for and what is happening.
Can you elaborate in what this does and whether it makes a difference if it's there or not?
I mean this code:
onchange="var qty_field = document.getElementById('hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>'); if (qty_field){<?php echo $input; ?>}"
This is the code which I didn't touch:
<?php if($this->params->get('show_delete',1)){ ?>
                  <div class="hikashop_cart_product_quantity_delete">
                    <a href="<?php echo hikashop_completeLink('product&task=updatecart&product_id='.$row->product_id.'&quantity=0&return_url='.urlencode(base64_encode(urldecode($this->params->get('url'))))); ?>" onclick="var qty_field = document.getElementById('hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>'); if(qty_field){qty_field.value=0; <?php echo $input; ?> qty_field.form.submit();} return false;" title="<?php echo JText::_('HIKA_DELETE'); ?>">
                      <img src="<?php echo HIKASHOP_IMAGES . 'delete2.png';?>" border="0" alt="<?php echo JText::_('HIKA_DELETE'); ?>" />
                    </a>
                  </div>
                <?php } ?>
Thanks