Make billing address available to twig

The billing address is defined as a placeholder in the "form-element--stripe-payment.html.twig". You can see the div with "address-element" ID.
If you want to change its place, you can do it from the template, but you can't implement it separately, as it is called while creating the payment intent.

Regarding sending invoices by email, I think the best way is to manage it from the Stripe admin panel.