Hi michu,
Yes, you can build a large led matrix out of led strips. One led strip is used as one logical led block with three leds.
Yes, you can use a 12v power supply to rainbowduino to driver the led strips.
The rainbowduino is an open source project, http://code.google.com/p/rainbowduino/downloads/listyou can download some demo code, in which there’s one for driving led strips.
May it helpful to you, and feel free to ask any question!
-Icing