1) Yes, you will to fit a PCI Ethernet card. Old network cards might only cater for 10 rather then 100 Mbps capacity. New cards are only c £5.
2) Strictly speaking MoDems are not Ethernet; I don't have the web sites which shows what Ethernet ports/sockets look like, but just do a Google for RJ45 Ethernet.
3) No; once everything is setup, your desktop does not have to be on for your laptop to access the wireless network, just the wireless router and your NTL cable modem has to be on.