Will making someone a Teacher remove their Admin status?
No it shouldn't. Moodle evaluates the permissions like "does the user have at least one role that allows her to do this?". Having multiple roles assigned is not uncommon. Quite opposite really as each role should only have the relevant permissions set.
In your case, you may want to revoke the capability to receive notifications from all roles, and then create a new role like "Notifications recipient" and assign that to selected users. Note this is just general approach I though, I haven't checked into details if that applies in your case, too. Let us know how it went!