by default there are 'billing' and 'delivery' sections during the checkout. If in the cart there are only digital products the delivery step is skipped entirely and only the billing section for the regular invoicing is showed. For the first order the name is taken from the logged in user info, not the email address.
After the first registration/order of a certain user, all his data are saved in the customers database, so for all subsequent orders they will be auto filled and skipped too.
You can show/hide some fields for address and tax code using the checkout options in the component configuration (see image attached)
Other than that if you want to customize the billing form you have to setup a template override for the component.