It looks like you are storing the Cateogry text and not the ID to identify categories in the products table. That would explain both issues.
That would mean line 56 of your menu.php should probably be:
$_POST["menuwriter_products_Category_ID"] = $row_menuwriter_category["Category"];
and the link on line 53 should be:
Category=<?php echo $row_menuwriter_category["Category"]; ?>&Search=1