Is it possible to have CBSubs automatically confirm a user's email address when someone has gone through the payment process of CBSubs?
Yes, edit your plans and on the workflows plan set confirmation to "No" for it to override CB.
Another related question is, would it be possible for the registration process to log someone in?
With CBSubs in place, no, otherwise yes using CB Auto Actions you could do this.
Is this possible?
Yes, download the newly released CB Verify Email plugin. It adds a second input below email fields for them to resupply the email. Works the same as the password fields basically.