How would it be powered?
There would be a motor and possibly a gear box. The gear box is optional. It would be powered by a 12 volt battery but there is potential for solar power to be used as this would have less environmental impact that a battery operated motor.
How would it change direction?
It order to make the pedalo go backwards and forwards you need to change the polarity. The polarity is the direction in which the motor spins. Another option is a control box which allows it to move in different directions, a DPDT switch would allow for moving forward, backward and turning it off.
How would it change speeds?
If you slow down the speed of the motor this increases the power (torque). The easiest and most reliable but most expensive way to do this is to use a gear box, this takes a fast sped and puts gears either side, the gears are different sizes and therefore different speeds. When an appropriate speed is chosen it would be fixed. Attached to the gear box is a shaft, if the paddle wheel was directly attached to the shaft the motor would be too close to the water, this would be extremely dangerous. To keep it away from the water you use a chain to transfer the power. The chain doesn't stretch although it does corrode. To prevent corrode you could use stainless steel but this would be expensive Another solution would be to use a rubber belt with reinforcement strips inside it although it must be under tension. The down side of this is that there would be a lack of friction which would stop the belt turning. The gear box must be designed to bring the speed down to an appropriate level, to do this you need to go from small clogs to big clogs, the bigger clog/sprocket has more teeth.