Is my "Netgear" modem supposed to stay solid green even if the computer is off?
Depends on the modem & its status light are supposed to work.
Assuming you mean some kind of broadband modem, your modem and your computer are two separate devices. The modem will continue to receive signals from your ISP even if there's no computer on the other end, as long as it's plugged into power and a phone/cable/fiber-optic line.
Depends. Each light on your ROUTER has a different meaning/function. Which light or lights are you talking about? Also, mention the model you're using. It helps to know what your device looks like.
yes, nnine stays green when its off too