Automatic Emails to New Users Upon Registration

I just spent quite a bit of time searching for the answer to this issue, but the existing forum posts seem to be from previous WP versions, in which the registration php is deprecated. So, …… here goes:

Using v.3.3.1. When new users register on my site, they are suppose to get an email with a starting password.
Problem 1: It appears not everyone is getting a password.
Problem 2: I would like to edit the email they receive.

Need solutions. Which files control these issues? In which file are the functions to be found? Where can I edit that email??

Thanks!