I just started skimmed over the Facebook API and OpenSocial API. In OpenSocial there is the construct of a gadget that is hosted on the container. You just write the code inside some XML structure and upload somehow to the container. So the code ist hosted by the OpenSocial provider. I couldn't find a pendent in the Facebook API. It seems that apps on Facebook must always be hosted on a developers system (and not on Facebook). Is that correct? Or did I miss something?
Yes, that's correct. Facebook is busy enough running its own stuff without your stuff keeping their servers busy!