Wow, this is a very interesting project. Here are a few thoughts that immediately popped into my mind.
1) Optics - I would try the Elcan Digital Hunter (approx $1200.00 I think). These have a 'video out' port. I've seen a lot of 'in car DVD players' at Costco that come with a pair of screens that hang over the front seats for the kids to watch. They are rather inexpensive. Mount the screen on the arm opposite the gun. The scope would be more clear and more accurate then a video camera. Probably more recoil friendly, and less bulky as well. A little pricy too, but your cousin would be able to capture his kills on video through the scope!
2) The Mount - Recoil - As was mentioned earlier, use shock absorbers. I keep thinking of artillery pieces, tank guns, and machinegun mounts.
3) The Mount - Movement - I would keep the mount pointed mostly forward to keep most the recoil rearwards. For small lateral and elevation movement (40 degrees maybe?), I would look into small servo motors attached to power screws.
I friend of mine is a remote controlled airplane nut. He's showed me a lot of the hobby sites that he gets stuff from. If you used the board from a plane remote control you could create a joystick to operate it similar to whatever he uses for his chair. He could line the chair up, then use the second control to adjust the rifle and take the shot. The servo motors allow for precision movement, as they are designed to control flaps and stuff on the planes. They were relatively inexpensive. Would this work given the specifics of your cousin's disability?
4)The Mount - Chair - I would see about making a precise fixture that can mount on his chair. If you needed to make a seperate fixture for each chair, I'd do it. Design them so that the rest of the mount would attach to either. (think of an aimpoint, and the dozen different mounting options you have for it).
5) Trigger - I like the solenoid idea. There would be several buttons on the airplane remote, one could be used for this. As far as safety, maybe you could put a delay on it (ie - the button will fire after being depressed for 3 seconds straight). So inadvertant taps will not set the rifle off. Maybe have another switch on the control give your cousin control of cocking the rifle, so that he could chamber the round on his own. Also, another switch could operate the rifle's safety.
The radio control keeps the wires down, though you would probably want the trigger wired in. As for the design of the switches, etc - these would be specific to your cousin's capabilities.
This may seem like a lot, but it would give him almost all control over his shot; his kill.
TM