>>> Am I correct in saying that a wireless router
>>> with an ethernet connection can be connected
>>> between the modem and the desktop
Generally "No" unless, unusually, you happen to be using an Ethernet ADSL Modem.
If you're currently using a USB ADSL Modem with the Desktop PC, this would be replaced with a *combined* wireless Router & ADSL Modem. The computers would then both connect to this either by ethernet cable or "wirelessly".
Does the Desktop PC actually have a "wired" network adapter installed? If so, it will appear in the Network Connections folder as the "Local Area Connection".
The 1394 adapter is *Firewire* and isn't suitable for connecting to a router.