Is an invoice attached to the order confirmation email, and does it use my invoice template?

Yes. The order confirmation email has an invoice PDF attached, named with the order ID.

That PDF uses a standard layout made from the order: your store name, the items, the coupon, shipping, tax and total. It does not use the template you set under Settings → Store Settings → Invoice Templates. Those templates are only used when you click Print invoice on an order in the admin.

If the PDF can't be created for some reason, the email is still sent without it. If you want to send a customer your branded invoice, open the order, click Print invoice and save it as PDF from the print window.