A:
Depends on what you mean by "turn off".
If you mean "unplug from electricity", the Ethernet ports won't work, because the modem is off. It needs power to run. If by "turn off" you mean "there are no LEDs on the Modem when I turn it off", yeah, it's dead.
There are some models that can be powered by the Ethernet cords themselves, but that requires Power over Ethernet support, and the devices connected to the modem to be giving Power over Ethernet... you likely don't have that in your home.
If you mean "turn off the Wireless broadcast from the menu," Wireless and Ethernet work independently. Ethernet will work even if Wireless is turned off, generally there is no way to disable ethernet ports on most household-grade modems.