I need to apply a 16% tax on all products, but I want the prices of products to include that tax everywhere: in the product list, description, checkout, etc., just as if the tax was a part of the original price. So far I only see that you can add the tax on checkout, as an extra, but I don't want the client to bother to even know that the tax exists. However, internally I want it to be separate from the product price.
How do I do it?
What you need to do is set up a Product Tax Class and apply it to all products, then set up a Tax rate & zones (16% and all zones), and then associate it all together in a tax rule so that it will apply the 16% rate to the Product tax class you created.
If you take a look at the Tax section in the documentation I'm sure it will also shed some more light for you: http://docs.magento.com/m1/ce/user_guide/tax/taxes.html
Next you will need to update the tax configuration and price display settings to match what you have entered under System > Configuration > SALES > Tax (E.g. you want to Display Product Prices in Catalog, Cart and Checkout as Including tax with no taz breakdown)
You can read more about this here: http://docs.magento.com/m1/ce/user_guide/configuration/sales/tax.html
I hope this helps!!