What happens to products that are no longer active or available on Amazon? How do I know they are being removed or syncing properly?

The synchronisation module will set Out-of-stock products' price to zero (no price). When a product will no longer be available on Amazon, the product will be removed from your website too.

You may easily supervise this, using the new module that we've introduced in the latest version of the plugin: http://docs.aa-team.com/woocommerce-amazon-affiliates/documentation/woozone-report/