The assembly instructions states that: Output 12V CC can be used as a power supply for other kits. Does this mean I could power a Arduino Mega with those 12V?
The ULN2803 type transistor array allows you to control the relays directly from a TTL or CMOS. For controlling the relay with open collector control, don't mount the transistor arrays. How does this apply to the arduino?
And last but not least, I can not figure out if I will be able to control this with my arduino digital outs. Is there enough power there to switch the relays? And do I need something between the arduino and the board?
You interface with the inputs of the ULN2803. This IC has been created just for the task of driving high current devices with low power outputs. You can connect the Arduino ports directly to them. The inputs consume below 1.5mA.
One important thing though is that you don't power the relay board from the +5V of your Arduino, that'll probably fry your Arduino. The board needs to be powered directly from a suitable power supply (which also can power the Arduino if necessary) and you need to make sure that the ground lines of your relay board (on the primary side) is connected to the ground pin of the Arduino.