First off, why are you using Net::SMTP::TLS rather than Net::SMTP? Do you need the security? If not then you might switch to Net::SMTP and follow the example here.

The HELLO is part of the authentication process, but I have not used it so I can't tell you any more than that, except that $EMAIL_HELLO is supposed to be a host.domain, not an email address.

Finally it's sending to me@mydomain.com because that's what you've got in the mail() method. Check the docs.