Chickens have been taught to look at a colour card then go and press the same colour button on a row of switches, so teaching them to open the door is possible, you'd need to reward them with something tasty.
Speaking of car electric window winders, some more fancy cars have an 'auto' button on the up/down switch for the window, I had a toyota with one, it was a self reseting circuit breaker, the button stayed down (or up) until the window hit the bottom (or top) at which point the current went up operating the 'breaker and the switch popped back into the central off position.
One thing about using a window winder, if you use a piece of wood the same size as the window, only square instead of car window shape, you'd be able to use most of the cars window mechanism to do the job simplifying things, if you use limit switches just drill holes in the wood to line up with up and down positions.
Heres a page explaining car relays, you might be able to make sense of the drawing above with this:
http://www.reuk.co.uk/Automotive-Relays.htm