Can I change the wording of the emails my store sends to customers?

Only partly, for now. Settings → Web Settings → Email Templates lists every template, and you can click Edit to change the Subject and Email body, then Save changes. But today only one of them is actually used when your store sends email: New Login Credentials — the email a customer gets when you add them from the admin, when you email them new login details, or when a guest checkout creates their account.

Order confirmation, shipping, refund and the other store emails are sent with fixed wording, so editing those templates (or setting them Inactive) does not change what customers receive yet.

Every email is sent in your store's branded email layout, and you can choose who gets copies, using Cc / Bcc on Settings → Web Settings → Email Provider.