I did see that you did have it plugged into other machines.
Regardless of whether Arduino is on the computer or not, it should still try to install a driver.
I was finding that despite the computer not talking, it did show a USB device plugged in on the task bar, and the UNO was in device manager.
In device manager you can choose to show hidden items under the view menu.
The other thing is if you click on each of the USB Root Hub icons, and click the power tab, it should show 100mA assigned to the device
I'm assuming that you have an USB icon on the task bar (when its running), so if you were to try applying sideways/up/down pressure to the USB cable at the socket end, it may show up as disconnected/connected to at least isolate a faulty socket/joint.
BTW you can run Arduino straight from the memory stick, it doesn't need to be installed (except the driver).
Portmon is not part of the operating system, you can download it from http://technet.microsoft.com/en-us/sysinternals/bb896644
or the whole suite of utilities from http://technet.microsoft.com/en-us/sysinternals/bb842062
(they do interlink on the page).
Procmon (in the suite) is something that shows what process is running, but be warned 10mins generates some 60-75,000 items.
The fact you tried it on a number of other computers without it asking for the driver, still makes me think you have a board issue.
However its going at the moment, so enjoy, and maybe let your supplier know.
We all have to start somewhere, but the internal workings of Windows wouldn't be the first place I would have picked.