The first section is a solution to the Unknown column problem.
In 'admin/categories.php', search for this line:
'products_cost_price' => tep_db_prepare_input($HTTP_POST_VARS['products_price_cost']),
Exchange that line with:
'products_price_cost' => tep_db_prepare_input($HTTP_POST_VARS['products_price_cost']),
This is for that other problem.
And in the same file, search for this:
. TEXT_PRODUCT_INVENTORY_COST .
Exchange it with:
. TEXT_PRODUCTS_PRICE_COST .
That should solve it.