No, you wouldn't need to leave one pc always on. Cable plugs into modem, modem plugs into router, router connects to pcs. You'd only need an always-on pc if you didn't have a router and were using a pc to perform the router functions instead.
Incidentally, separate modem/router is nice, in that if you ever switch to DSL you can still use the router. Other than saving a little space, I'm not aware of any compelling advantage to a combo modem/router.