This is a really cool idea. If you really want to do it, the folks on this board will totally help you on it. Let me start you off with a few questions that matter a lot.
How far is the coup from power? Do you have something like a light to scare off the 'coons nearby?
How far is the coup from where some computer stuff is plugged in. Like, do you have an ethernet plug somewhere nearby? How hard would it be to run some wires to it.
Sunlight? Maybe solar power is possible to recharge batteries.
See, the feeder has to be with the chickens, but the arduino can be somewhere else. Suppose you built a screw driven conveyor that dropped corn and ran wires from it to the house? If that's too hard or not fun enough, how about an XBee on the feeder with a little control circuitry that is wireless to an Arduino in the house. If that isn't fun enough, put the arduino in the coup with the feeder hook an XBee to it and another one in the house hooked to something on the web.
Webcam? You simply got to have a webcam ! We would all want to see how fat we could make your chickens.
Think about the physical aspects of doing this. The electronics will be based on how you think it could be done the best.