When I power on my MacBook Pro, will it already have AirPort Utility installed (and will it recognize the AirPort Extreme)
Yes ... and yes.
...and is there anyway to keep the modem connected to both the desktop and the AirPort Extreme?
Yes. You can disconnect the Ethernet cable going from the modem at the PC and connect it to the WAN (circle of dots) port on the 802.11n AirPort Extreme Base Station (AEBSn). Then you can connect a second Ethernet cable between one of the available LAN (opposing arrows) port on the AEBSn and the PC. However, before doing so I would recommend that you perform a complete power recycle of your networking equipment to make sure that the modem and the new AEBSn "talk" with each other.
My desktop says it has an 802.11n Wireless LAN Card if that's beneficial.
Since your desktop is Wi-Fi ready, you can also elect to connect the desktop to the AEBSn by wireless instead of by Ethernet.