I think most people here leave their routers on 24/7.
Particularly if you are on maxdsl and once your line is stable, it can help keep the IPprofile stable... but at the end of the day its a matter of choice if you turn it off or not.
If, as it sounds, that your ISP has advised you to keep the router switched on, then yes you will have to plug the router into a different electrical socket and bypass the energy saving socket. As soon as the adapter senses that the PC is turned off then everything else plugged into that adapter will power down.
Hmmm actually just thinking about it.. Im wondering how it powers down.. and if the router has chance to send the "dying gasp" signal to the DLM.
Its probably NOT a good idea to have a router attached to one of these adapters if youre on Maxdsl, because the router wont go through a proper shutdown proceedure, and could dupe the DLM into thinking youve lost sync for other reasons.