Hi,
1. I checked your website but what you have on your screenshot is not the next button but the registration button. So it's normal that it doesn't change anything. The next button on the first step of your checkout is already on the right.
So for the button of your screenshot, it should be that CSS that you want to add:
#hikashop_register_form_button{ float:right; }
2. In that case, instead of the modification proposed by Xavier, you can change the code:
<input id="hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>" type="text" name="item[<?php echo $row->cart_product_id;?>]" class="hikashop_product_quantity_field" value="<?php echo $row->cart_product_quantity; ?>" onchange="var qty_field = document.getElementById('hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>'); if (qty_field){<?php echo $input; ?>}; return true;" />
by:
<input id="hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>" type="hidden" name="item[<?php echo $row->cart_product_id;?>]" value="<?php echo $row->cart_product_quantity; ?>" /><?php echo $row->cart_product_quantity; ?>
and remove the code:
<div class="hikashop_cart_product_quantity_refresh">
<a class="hikashop_no_print" href="#" onclick="var qty_field = document.getElementById('hikashop_checkout_quantity_<?php echo $row->cart_product_id;?>'); if (qty_field && qty_field.value != '<?php echo $row->cart_product_quantity; ?>'){<?php echo $input; ?> qty_field.form.submit(); } return false;" title="<?php echo JText::_('HIKA_REFRESH'); ?>">
<img src="<?php echo HIKASHOP_IMAGES . 'refresh.png';?>" border="0" alt="<?php echo JText::_('HIKA_REFRESH'); ?>" />
</a>
</div>