Gmail – “550 550 Authentication is required for relay (state 13)”

make sure you're the "owner" of the domain you created. Sometimes this changes things a bit. I was an admin for a site that I was writing a script for, then one day I was no longer listed as an owner of the site and couldn't access google apps script. This might affect your mail account at that domain.

MX records control only the server direction (i.e. messages sent to @example.com will be handled by the server located at 123.123.123.123). It's the local mail server that decides that admin@example.com and mail@example.com correspond to the same inbox. Google uses the term nickname for forwarded addresses.

In order to create a nickname for an existing user/inbox in Google Apps, you need to do the following:

1. Go to your Google Apps admin dashboard
2. Click on Organization and users in the blue navigation menu on top of the page
3. Click on the user, to which you want to assign a nickname
4. In the User information tab (the default one) click on Add a nickname (it's right below Email quota)