So just to be sure, you have the router, then the switch, then the devices all connected to the switch? If not, connect it this way first, making sure there is only one connection from router to switch.
Next, is it always the same PC that drops off? Can you swap the Ethernet cables from the working PC to the other to see if this resolves it? It could be a faulty Ethernet cable or network card.
I'd make everything DHCP to start to ensure the router is issuing IP's.