I think you would be well advised to read a little about home networks, especially the network infrastructure devices.
A couple of basics: You should avoid ICS if possible (IMO). It is too much trouble. A router will allow all of your computers, etc., access to the internet (WAN) as well as manage the IP assignment and related chores for your home network. Plus, they are cheap.
To answer some of your questions, yes, the ICS computer needs to be on whenever anyone wants access to the internet. Is that 24/7? Depends on your needs.
You can share a printer a variety of ways. You can simply enable printer sharing on the computer the printer is attached to. This requires that that computer be powered up whenever anyone wants to use the printer.
You can share the printer using a USB switch. This requires the computers sharing the printer to be relatively close to the printer (USB cable length) and sometimes (depending on the switch) requires manual switching. There are limits as to the number of computers than can share directly (usually 2).
You can combine the two above techniques and allow other computers on the network to share the printer attached to the USB switch. This requires one of the computers physically attached to the printer to be powered on and have printer sharing enabled.
You can purchase a printer server that will attach to ethernet. This will allow all computers on your network to share the printer independently. No other computer needs to be powered on.