for the error; Field 'ProductWeight' doesn't have a default value
In the database, edit the products table, either assign a default value, or set the ProductWeight column to allow NULL Values.
for the headers already sent error, change lines 85 - 87:
see the following post for more details on that error:
For the product categories, lets back up a moment.
You assign the product to a category on the products insert page
read my initial response about that again:
Run the data assist wizard, and select the products table. Set category column to be a menu, click the lightning bolt next to the menu and select the categories table. Set the category ID to be used as the value, and the Category name to be used as the label.
re read this part:
this will add a select list to the products insert page that will allow you to select the category that the product belongs to.
To be able to add other categories, you will need to re run the data assist wizard for the categories table, but that is not the same as assigning the product to a category.