As you have a newer Mac, I am going to assume that you are using the term Airport, to mean an Apple Airport Exteme Basestation. Correct me if I'm wrong.
To connect both machines to the internet, at the same time, try this.
The providers cable, or phone line for DSL, should be connected to the providers supplied (usually) modem.
The modem, in a single user setup, is directly connected to the desktop via a length of Cat5 ethernet cable. This is a lot fatter than a phone cable and has a 6 pin connector (male) on both ends. They look like regular phone connectors on steroids. The are bigger.
This cable plugs in to the network port of the desktop.
Your setup should be:
1. Providers cable into the modem
2. Cat5 cable from Modem to Apple AIrport base station. (into the WAN socket at the rear)
3. Laptop connects wirelessly to the Airport Base station
4. Desktop connects directly to the Airport base station, with a length of Cat5 ethernet cable, to one of the four empty ports at the back of the Airport base station.
5. Both go surfing together.
6. There is no #6
If you do not have an Apple Airport base station, the above works with any wireless router. They all, almost, have 1 Wide Area Network (WAN) port and 4 Local Area Network (LAN) ports.
That piece of cable that you keep switching around? That's Cat5 ethernet cable. Now you will recognize it again in the store. Of course, the label on the package will give it away too!
Let us know how you get on
Once we have that sorted out, we can try something for the blue screen