Yes, your WooCommerce site requires hosting if you want it to be accessible on the internet. You may work on your WooCommerce on your local computer without hosting, but if you wish to publish it live, then you need to sign up for hosting that supports WooCommerce.

WooCommerce is a plugin for the CMS called WordPress. If you wish to build an online shop, you would get both WordPress and WooCommerce installed for a start. That means that essentially the hosting requirements for servicing both of these softwares are the same. The difference is in the quantity of the server resources that may be needed to ensure the proper functioning of a site built on WordPress vs. one built on WooCommerce. Usually, online shops built on WooCommerce use more server resources than websites built on WordPress and hence, hosting WooCommerce may be more expensive. Of course, it all depends on how big your site or e-shop is and especially how big their databases are. The bigger the sites, the heavier resource users they become.

No, WooCommerce as a stand-alone application does not impose any fees. The software itself is free for anyone to use and there are no transaction or other fees associated. Any costs related to your WooCommerce sites depend on how you build, configure and manage it.