Does WooCommerce Amazon Fulfillment charge fees for shipping or handle the collection of payment at checkout?
No. WooCommerce Amazon Fulfillment has nothing to do with collecting payment from the customer or determining shipping options (these are both 100% handled by WooCommerce). WooCommerce Amazon Fulfillment's only job is to move raw data from WooCommerce to FBA.
However, it's important to understand that Amazon DOES charge you various fees for shipping. What FBA bills you for the fulfillment service has nothing to do with what you charge your customers for shipping -- although we recommend you to configure your shipping options in WooCommerce in order to cover your FBA shipping costs. WooCommerce Amazon Fulfillment just so happens to let you map your WooCommerce shipping methods to FBA "shipping speeds". Read our article titled "Supported Shipping Methods and Options" for more information on how to set that up.