No emails are going out

I thought they were working, but now I am not sure. Now not one email is being generated.

In particular the one with the person's new membership number and temporary password, sent automatically once they were marked paid, enabled, given a number, and approved.

Try sending an email from CB > User Management to your self by checking your users checkbox then clicking Mass Mailer. You may also want to check with your host to see if emails are being stopped in the mailer server for whatever reason if mass mailer is working fine.