I noticed awhile back in all of my Gmail account settings something odd.
Was curious if anyone else has noticed this before?
EDIT: I am referring to the firstname.lastname@example.org account
Nope, your account may have been hacked into and that innocent-looking address was used to get a copy of all your emails. The @googlemail.com domain is interchangeable with @gmail.com, meaning that someone could have easily registered that address as an @gmail.com address. In other words, this address by no means is some official Google email address.
I would delete that account as a forwarding address, and assume that any email you received, as long as that setting was enabled, may have been read by a third party. The forwarding could have been setup in the settings page of your screenshot (it shows it as off in the screenshot, but maybe it was enabled at a previous time), or in a mail filter.
It might be some kind of service that you signed up for, which does this automatically for you. However searching on Google for that email address doesn't bring up anything, so I doubt that's the case. This sounds similar to a situation described in this question.
Yes. If you want to automatically forward email between accounts, or use Microsoft Outlook, Windows Mail or Apple Mail, you would want to check these settings.