And yet, when I use the CB User Manager to add a new user, an email is still sent to the user, and the password I specify is replaced by a random password.
User will always receive either Pending or Welcome email, regardless if created from backend or not. However they should receive a random password if you're not using random passwords. Please ensure they're not just receiving the encrypted password. Random passwords should be no greater then I believe 8-12 characters.
How do I *prevent* all emails from being sent when users are registered from the back-end?
You'll have to empty the Subject and Body of the Welcome/Pending emails within CB > Configuration > Registration.
PS: I also have "Ajax Username checker" = no, and yet a realtime username is done at the back-end.
Ajax checks will always function in backend regardless of configuration.