Thanks for the post and pictures of the strip. Very informative. It is interesting that they are using 12V rather than 5V that the 8806 needs. Looking at the resistors and layout, it appears that this strip will have limitations.
You can use the arduino for the CLK and DA pins and then use your 12V power source to power it. The LPD8806 library should work with this strip.
I think with this configuration, you will only be able to control colors every 3rd LED rather than every LED as with the strips on Adafruit. The LEDs on each side of the 8806 appear to be in series with an additional resistor for each to drop voltage further. The max power dissipation of the 8806 is 600mW so with 6 RGB LEDs, it would be able to supply a maximum of only 6.7mA to each leg of each LED. This coincides with the 180 ohm resistors for the green and blue legs in the pics. These resistors would limit current to around 6mA for each color. This would mean it would not be as bright as the strip on Adafruit that can supply the full rated 20mA to each leg of each LED.
But this is all based on looking at the photos and my limited knowledge of these things. You won't really know until you hook it up and find out. 
Please post a follow up on this.