For the robot arm I would check out lynxmotion.com They have a nice kit but it may be a bit out of your price range.
Otherwise servocity.com and sparkfun electronics and Trossen Robotics are all great sites. And thats only four off the top of my head.
For the 9.6v battery, any 9.6v RC car battery would work great. Visit a local hobby shop for those, and if there is no limit get the ones with the highest mAh for the longest run time.
For all my RC stuff I would either visit a local hobby shop or towerhobbies.com They usually have cheap radios that can be used and sometimes have deals where you buy a radio and get the receiver and servos for free.
For the wiring, it all depends on what you get and what you want to add later on but if you know the theory of electronics then this should not be an issue for you as most of it is plug and play anyways.
If you only got 3 weeks, then you better get cracking and divide up your time between researching parts, ordering parts, prototyping, building, and lastly testing. Sounds like you only got about 3 days to do each step, and that's assuming parts arrive in order and on time.
Good luck and if you have more specific questions, feel free to ask.
- Will
