Email forwarders send a copy of emails and keep an original. You can setup multiple forwarders to and/or from the same account.
I have found that if you send email from Gmail to an email address which forwards back to the same Gmail, then Gmail will not receive that forward. (Host World is the same way.)
How can I stop my Host World email from keeping a copy?
You must delete the mailbox from Email Accounts, and only keep the forwarder. This way the email will still forward, but the emails will not be saved at the first address.
Can I have all the outgoing email forwarded to my email address?
No, this can only be done on a dedicated server.
Is it possible to set up e-mail accounts that forward to more than one e-mail address?
Yes you can. You just need to log into your site's cPanel and click the Forwarders icon, and then add a forwarder with the destinations separated by commas like so:
firstname.lastname@example.org --> email@example.com,firstname.lastname@example.org,email@example.com