I've managed to acquire a Google Nexus One, that had an unfortunate encounter with a car door. The screen still accepts touch input, but the LCD itself is broken - no display at all. What is something "cool" I can do with it? My main concern is, is it possible to root it and get a remote shell without being able to see the screen?
You can get a remote shell with it, without the screen working. You will have to install the android SDK on your PC.
Another thing you can do is use Android ScreenCast to view the screen remotely. It doesn't have the best frame rate, but at least you will be able to interact with the screen. You may have to use this to get the remote shell working, as you may have to turn on USB Debugging in Settings -> Applications -> Development -> USB Debugging
You could use it to test accessibility applications that are used by blind or visually impaired; these apps rely on audio and haptic feedback to tell you what is going on. Not sure how you'd get into that mode, however.
You can use it as wifi for all family members, buy a sim with good data plan and use it in broken phone and plug it in for charging and turn its wifi hot spot which can be used by other family member.
Just go to a store and say that it was an incident.. Pay the fee and get a new display